%%% -*-BibTeX-*- %%% ==================================================================== %%% BibTeX-file{ %%% author = "Nelson H. F. Beebe", %%% version = "1.24", %%% date = "26 October 2023", %%% time = "10:12:54 MDT", %%% filename = "agm.bib", %%% address = "University of Utah %%% Department of Mathematics, 110 LCB %%% 155 S 1400 E RM 233 %%% Salt Lake City, UT 84112-0090 %%% USA", %%% telephone = "+1 801 581 5254", %%% FAX = "+1 801 581 4148", %%% URL = "https://www.math.utah.edu/~beebe", %%% checksum = "58433 13217 56002 563890", %%% email = "beebe at math.utah.edu, beebe at acm.org, %%% beebe at computer.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "arithmetic--geometric mean; bibliography; %%% BibTeX; geometric--arithmetic mean", %%% license = "public domain", %%% supported = "yes", %%% docstring = "This is a bibliography of publications about %%% the arithmetic--geometric mean (AGM) %%% iteration, discovered by Lagrange before %%% 1785, and independently by Carl Friedrich %%% Gauss (1777--1855) in 1799. Adrien-Marie %%% Legendre (1752--1833) discovered a relation %%% between certain elliptic integrals that led %%% Gauss to apply the AGM to the calculation of %%% the mathematical constant pi. %%% %%% However, the AGM's first well-known %%% applications, to the high-precision %%% computation of pi, and of high-precision %%% computation of elementary and elliptic %%% functions, were not published until 1971 and %%% 1976 (see entries Carlson:1971:AIA, %%% Brent:1976:FMP, and Salamin:1976:CUA). There %%% was earlier work in the 1920s (see the books %%% and papers by Louis V. King) using the AGM to %%% compute Jacobian elliptic functions, but %%% King's publications seem not to have been %%% widely known or appreciated. %%% %%% Gauss' work is better known, and the AGM is %%% often (improperly) credited to him alone; see %%% entry Gauss:1992:AGM for a Spanish %%% translation of his original work, from the %%% first publication in the original Latin in %%% his collected works (entry Werke Vol. X-1 %%% (1917)). %%% %%% The AGM provides an iterative process for %%% computing various constants and functions, %%% often with quadratric convergence. Unlike %%% Newton--Raphson iteration, the AGM process is %%% not self correcting: errors accumulate with %%% each iteration. That is a disadvantage with %%% hardware arithmetic of fixed precision, but %%% is easily handled in software %%% multiple-precision arithmetic by computing %%% with a few more digits than are eventually %%% needed. The AGM iteration is comparatively %%% simple to program, and especially for %%% arbitrary-precision arithmetic, provides a %%% convenient way to compute several important %%% constants and functions. %%% %%% Since the mid-1990s, in a few cases, variants %%% of the AGM have been discovered possessing %%% convergence of order 3, 4, 5, ..., 9. For %%% example, pi can be calculated to a thousand %%% million digits with only ten 9-th order AGM %%% iterations. %%% %%% At version 1.24, the year coverage looked %%% like this: %%% %%% 1866 ( 1) 1919 ( 0) 1972 ( 4) %%% 1868 ( 1) 1921 ( 1) 1974 ( 0) %%% 1869 ( 0) 1922 ( 0) 1975 ( 1) %%% 1870 ( 0) 1923 ( 0) 1976 ( 7) %%% 1871 ( 0) 1924 ( 1) 1977 ( 2) %%% 1872 ( 0) 1925 ( 0) 1978 ( 5) %%% 1873 ( 0) 1926 ( 0) 1979 ( 3) %%% 1874 ( 0) 1927 ( 1) 1980 ( 1) %%% 1875 ( 0) 1928 ( 1) 1981 ( 3) %%% 1876 ( 0) 1929 ( 0) 1982 ( 2) %%% 1877 ( 0) 1930 ( 0) 1983 ( 3) %%% 1878 ( 0) 1931 ( 0) 1984 ( 6) %%% 1879 ( 0) 1932 ( 0) 1985 ( 3) %%% 1880 ( 0) 1933 ( 0) 1986 ( 5) %%% 1881 ( 0) 1934 ( 0) 1987 ( 10) %%% 1882 ( 0) 1935 ( 0) 1988 ( 22) %%% 1883 ( 0) 1936 ( 1) 1989 ( 8) %%% 1884 ( 0) 1937 ( 0) 1990 ( 9) %%% 1885 ( 0) 1938 ( 0) 1991 ( 8) %%% 1886 ( 0) 1939 ( 0) 1992 ( 9) %%% 1887 ( 0) 1940 ( 0) 1993 ( 13) %%% 1888 ( 0) 1941 ( 0) 1994 ( 12) %%% 1889 ( 0) 1942 ( 1) 1995 ( 12) %%% 1890 ( 0) 1943 ( 0) 1996 ( 15) %%% 1891 ( 0) 1944 ( 1) 1997 ( 17) %%% 1892 ( 0) 1945 ( 1) 1998 ( 8) %%% 1893 ( 0) 1946 ( 1) 1999 ( 13) %%% 1894 ( 0) 1947 ( 0) 2000 ( 12) %%% 1895 ( 0) 1948 ( 0) 2001 ( 6) %%% 1896 ( 0) 1949 ( 0) 2002 ( 6) %%% 1897 ( 0) 1950 ( 0) 2003 ( 14) %%% 1898 ( 0) 1951 ( 0) 2004 ( 11) %%% 1899 ( 0) 1952 ( 0) 2005 ( 2) %%% 1900 ( 0) 1953 ( 0) 2006 ( 9) %%% 1901 ( 0) 1954 ( 0) 2007 ( 9) %%% 1902 ( 0) 1955 ( 0) 2008 ( 11) %%% 1903 ( 0) 1956 ( 3) 2009 ( 9) %%% 1904 ( 0) 1957 ( 0) 2010 ( 13) %%% 1905 ( 0) 1958 ( 1) 2011 ( 19) %%% 1906 ( 0) 1959 ( 0) 2012 ( 20) %%% 1907 ( 0) 1960 ( 1) 2013 ( 16) %%% 1908 ( 0) 1961 ( 0) 2014 ( 16) %%% 1909 ( 0) 1962 ( 0) 2015 ( 15) %%% 1910 ( 0) 1963 ( 3) 2016 ( 26) %%% 1911 ( 1) 1964 ( 0) 2017 ( 13) %%% 1912 ( 0) 1965 ( 3) 2018 ( 3) %%% 1913 ( 0) 1966 ( 3) 2019 ( 1) %%% 1914 ( 0) 1967 ( 3) 2020 ( 6) %%% 1915 ( 0) 1968 ( 4) 2021 ( 0) %%% 1916 ( 0) 1969 ( 1) 2022 ( 1) %%% 1917 ( 1) 1970 ( 4) 2023 ( 1) %%% 1918 ( 0) 1971 ( 6) %%% %%% Article: 390 %%% Book: 15 %%% InCollection: 36 %%% InProceedings: 7 %%% Misc: 2 %%% PhdThesis: 2 %%% Proceedings: 7 %%% TechReport: 7 %%% Unpublished: 19 %%% %%% Total entries: 485 %%% %%% Data for this entry have been collected from %%% the BibNet Project and TeX User Group %%% bibliography archives, and the arXiv.org, %%% Elsevier ScienceDirect, MathSciNet, Springer %%% Link, and Wiley Online databases. %%% %%% In this bibliography, entries are sorted %%% first by ascending year, and within each %%% year, alphabetically by author or editor, %%% and then, if necessary, by the 3-letter %%% abbreviation at the end of the BibTeX %%% citation tag, using the bibsort -byyear %%% utility. Year order has been chosen to %%% make it easier to identify the most recent %%% work. %%% %%% The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ====================================================================

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%%% ==================================================================== %%% Institutional abbreviations:

@String{inst-CECM= "Centre for Experimental and Constructive Mathematics (CECM) at Simon Fraser University (SFU)"} @String{inst-CECM:adr= "Burnaby, BC V5A 1S6, Canada"}

%%% ==================================================================== %%% Journal abbreviations:

@String{j-ABSTR-APPL-ANAL= "Abstract and Applied Analysis"} @String{j-ACTA-GEOD-GEOPHYS-HU= "Acta Geodaetica et Geophysica Hungarica"} @String{j-ADV-APPL-STAT= "Advances and Applications in Statistics"} @String{j-ADV-COMPUT-MATH= "Advances in Computational Mathematics"} @String{j-ADV-MATH= "Advances in Mathematics"} @String{j-AEQUATIONES-MATHEMATICAE= "Aequationes Mathematicae"} @String{j-AMER-J-PHYSICS= "American Journal of Physics"} @String{j-AMER-MATH-MONTHLY= "American Mathematical Monthly"} @String{j-AMER-STAT= "The American Statistician"} @String{j-ANN-FUNCT-ANAL= "Annals of Functional Analysis"} @String{j-ANN-SC-NORM-SUPER-PISA-CL-SCI= "Annali della Scuola normale superiore di Pisa, Classe di scienze"} @String{j-APPL-MATH-COMP= "Applied Mathematics and Computation"} @String{j-APPL-MATH-LETT= "Applied Mathematics Letters"} @String{j-APPL-MATH-MODEL= "Applied Mathematical Modelling"} @String{j-APPL-MATH-SCI-RUSE= "Applied Mathematical Sciences (Ruse)"} @String{j-ARCH-INEQUAL-APPL= "Archives of Inequalities and Applications."} @String{j-AUSTRALIAN-MATH-SOC-GAZ= "Australian Mathematical Society Gazette"} @String{j-BOLL-STOR-SCI-MAT= "Bollettino di Storia delle Scienze Matematiche"} @String{j-BULL-AUSTRAL-MATH-SOC= "Bulletin of the Australian Mathematical Society"} @String{j-BULL-LOND-MATH-SOC= "Bulletin of the London Mathematical Society"} @String{j-C-R-ACAD-SCI-I= "Comptes rendus de l'Acad{\'e}mie des sciences. S{\'e}rie I, Math{\'e}matique"} @String{j-CAN-MATH-BULL= "Bulletin canadien de math\-{\'e}\-mat\-iques = Canadian Mathematical Bulletin"} @String{j-COLLEGE-MATH-J= "College Mathematics Journal"} @String{j-COMMUN-KOREAN-MATH-SOC= "Communications of the Korean Mathematical Society"} @String{j-COMPUT-MATH-APPL= "Computers and Mathematics with Applications"} @String{j-COMPUTERS-AND-GRAPHICS= "Computers and Graphics"} @String{j-CONG-NUM= "Congressus Numerantium"} @String{j-CONST-APPROX= "Constructive Approximation"} @String{j-CWI-QUARTERLY= "CWI Quarterly"} @String{j-DEUTSCH-MATH-V= "Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)"} @String{j-ECON-QUAL-CONTROL= "Economic Quality Control"} @String{j-ELEM-MATH= "Elemente der Mathematik"} @String{j-ENSEIGN-MATH-2= "L'Enseignement Math{\'e}matique. Revue Internationale. 2e S{\'e}rie"} @String{j-EUR-J-COMB= "European Journal of Combinatorics"} @String{j-EXP-MATH= "Experimental Mathematics"} @String{j-EXPO-MATH= "Expositiones Mathematicae"} @String{j-FIB-QUART= "Fibonacci Quarterly"} @String{j-FUNCT-SPACES= "Journal of Function Spaces"} @String{j-GAZ-MATH= "Gazette des Math{\'e}maticiens"} @String{j-HOKKAIDO-MATH-J= "Hokkaido Mathematical Journal"} @String{j-IMA-J-NUMER-ANAL= "IMA Journal of Numerical Analysis"} @String{j-INDIAN-J-MATH= "Indian Journal of Mathematics"} @String{j-INT-J-MATH= "International Journal of Mathematics"} @String{j-INT-J-MATH-EDU-SCI-TECH= "International Journal of Mathematical Education in Science and Technology"} @String{j-INT-J-MATH-MATH-SCI= "International Journal of Mathematics and Mathematical Sciences"} @String{j-INT-J-NUMBER-THEORY= "International Journal of Number Theory"} @String{j-INT-J-PROD-ECON= "International Journal of Production Economics"} @String{j-INT-J-PURE-APPL-MATH= "International Journal of Pure and Applied Mathematics"} @String{j-INT-J-SYST-SCI= "International Journal of Systems Science"} @String{j-INTERDISCIP-INFORM-SCI= "Interdisciplinary Information Sciences"} @String{j-IRISH-MATH-SOC-BULL= "Irish Mathematical Society Bulletin"} @String{j-ISRAEL-J-MATH= "Israel Journal of Mathematics"} @String{j-ITAL-J-PURE-APPL-MATH= "Ital. J. Pure Appl. Math."} @String{j-J-ACM= "Journal of the ACM"} @String{j-J-AM-STAT-ASSOC= "Journal of the American Statistical Association"} @String{j-J-APPL-MATH= "Journal of Applied Mathematics"} @String{j-J-APPROX-THEORY= "Journal of Approximation Theory"} @String{j-J-AUTOM-REASON= "Journal of Automated Reasoning"} @String{j-J-COMPUT-APPL-MATH= "Journal of Computational and Applied Mathematics"} @String{j-J-ELECTROST= "Journal of Electrostatics"} @String{j-J-FUNCT-ANAL= "Journal of functional analysis"} @String{j-J-INEQUAL-APPL= "Journal of Inequalities and Applications"} @String{j-J-LOND-MATH-SOC= "Journal of the London Mathematical Society"} @String{j-J-LOND-MATH-SOC-2= "Journal of the London Mathematical Society. Second Series"} @String{j-J-MATH-ANAL-APPL= "Journal of Mathematical Analysis and Applications"} @String{j-J-MATH-INEQUAL= "Journal of Mathematical Inequalities"} @String{j-J-MATH-PHYS= "Journal of Mathematical Physics"} @String{j-J-MATH-SCI-ADV-APPL= "Journal of Mathematical Sciences. Advances and Applications"} @String{j-J-NUMBER-THEORY= "Journal of Number Theory"} @String{j-J-THEOR-PROBAB= "Journal of Theoretical Probability"} @String{j-JIPAM-J-INEQUAL-PURE-APPL-MATH= "IPAM. Journal of Inequalities in Pure and Applied Mathematics"} @String{j-LECT-NOTES-COMP-SCI= "Lecture Notes in Computer Science"} @String{j-LECT-NOTES-MATH= "Lecture Notes in Mathematics"} @String{j-LIN-MULT-ALGEBRA= "Linear Multilinear Algebra"} @String{j-LIN-AND-MULT-ALGEBRA= "Linear and Multilinear Algebra"} @String{j-LINEAR-ALGEBRA-APPL= "Linear Algebra and its Applications"} @String{j-MAPLE-TECH-NEWS= "Maple Technical Newsletter"} @String{j-MATH-ANN= "Mathematische Annalen"} @String{j-MATH-COMPUT= "Mathematics of Computation"} @String{j-MATH-GAZ= "The Mathematical Gazette"} @String{j-MATH-INEQUAL-APPL= "Mathematical Inequalities \& Applications"} @String{j-MATH-INTEL= "The Mathematical Intelligencer"} @String{j-MATH-MAG= "Mathematics Magazine"} @String{j-MATH-NACHR= "Mathematische Nachrichten"} @String{j-MATH-PROC-CAMB-PHILOS-SOC= "Mathematical Proceedings of the Cambridge Philosophical Society"} @String{j-MATH-STUDENT= "The Mathematics Student"} @String{j-NAMS= "Notices of the American Mathematical Society"} @String{j-NIEUW-ARCHIEF-WISKUNDE-4= "Nieuw Archief voor Wiskunde. Vierde Serie"} @String{j-NORDISK-MATH-TIDSKR= "Nordisk Matematisk Tidskrift"} @String{j-NUM-MATH= "Numerische Mathematik"} @String{j-NUMER-FUNCT-ANAL-OPTIM= "Numerical Functional Analysis and Optimization"} @String{j-OBZORNIK-MAT-FIZ= "Obzornik Mat. Fiz."} @String{j-OCTOGON-MATH-MAG= "Octogon Mathematical Magazine"} @String{j-OPER-MATRICES= "Operators and Matrices"} @String{j-PAC-J-APPL-MATH= "Pacific Journal of Applied Mathematics"} @String{j-PAC-J-MATH= "Pacific Journal of Mathematics"} @String{j-PHYS-LET-A= "Physics Letters A"} @String{j-PI-MU-EPSILON-J= "Pi Mu Epsilon Journal"} @String{j-PROC-AM-MATH-SOC= "Proceedings of the American Mathematical Society"} @String{j-PROC-JAPAN-ACAD-SER-A-MATH-SCI= "Proceedings of the Japan Academy of Sciences. Series A. Mathematical Sciences"} @String{j-PROC-R-SOC-EDINB-SECT-A-MATH= "Proceedings of the Royal Society of Edinburgh. Section A, Mathematical and Physical Sciences"} @String{j-PROC-R-SOC-LOND-SER-A-MATH-PHYS= "Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character"} @String{j-PUBL-MATH-DEBRECEN= "Publicationes Mathematicae Debrecen"} @String{j-RAMANUJAN-J= "The {Ramanujan} Journal"} @String{j-REND-CIRC-MAT= "Rendiconti del Circolo matematico di Palermo"} @String{j-RESONANCE= "Resonance"} @String{j-ROCKY-MOUNTAIN-J-MATH= "Rocky Mountain Journal of Mathematics"} @String{j-SCI-MATH-JPN= "Scientiae Mathematicae Japonicae"} @String{j-SIAM-J-APPL-MATH= "SIAM Journal on Applied Mathematics"} @String{j-SIAM-J-MAT-ANA-APPL= "SIAM Journal on Matrix Analysis and Applications"} @String{j-SIAM-J-MATH-ANA= "SIAM Journal on Mathematical Analysis"} @String{j-SIAM-J-OPT= "SIAM Journal on Optimization"} @String{j-SIAM-REVIEW= "SIAM Review"} @String{j-STAT-PROB-LETT= "Statistics \& Probability Letters"} @String{j-STOCH-PROC-APPL= "Stochastic Processes and Their Applications"} @String{j-TAMKANG-J-MATH= "Tamkang Journal of Mathematics"} @String{j-TOPOLOGY= "Topology"} @String{j-TRANS-AM-MATH-SOC= "Transactions of the American Mathematical Society"} @String{j-TWO-YEAR-COLL-MATH-J= "Two-Year College Mathematics Journal"} @String{j-UTIL-MATH= "Utilitas Mathematica"} @String{j-Z-ANGE-MATH-MECH= "{Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik}"} @String{j-ZH-VYCHISL-MAT-MAT-FIZ= "Zhurnal Vychislitel'no{u{i}} Matematiki i Matematichesko{u{i}} Fiziki"}

%%% ==================================================================== %%% Publisher abbreviations:

@String{pub-ACADEMIC= "Academic Press"} @String{pub-ACADEMIC:adr= "New York, NY, USA"} @String{pub-CAMBRIDGE= "Cambridge University Press"} @String{pub-CAMBRIDGE:adr= "Cambridge, UK"} @String{pub-IEEE= "IEEE Computer Society Press"} @String{pub-IEEE:adr= "1109 Spring Street, Suite 300, Silver Spring, MD 20910, USA"} @String{pub-SV= "Spring{\-}er-Ver{\-}lag"} @String{pub-SV:adr= "Berlin, Germany~/ Heidelberg, Germany~/ London, UK~/ etc."} @String{pub-WILEY= "Wiley"} @String{pub-WILEY:adr= "New York, NY, USA"}

%%% ==================================================================== %%% BibTeX entries, sorted by year, and then by citation label:

@Book{Gauss:1866:W, author = "Carl Friedrich Gauss", title = "Werke", volume = "3", publisher = "Koniglichen Gesellschaft der Wissenschaften", address = "G{\"o}ttingen, Germany", pages = "????", year = "1866", bibdate = "Tue Mar 14 19:03:18 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, author-dates = "1777--1855", remark = "See pages 361--403 for AGM work.", } @InCollection{Lagrange:1868:X, author = "Joseph-Louis Lagrange", booktitle = "{\OE}uvres. ({French}) [{Works}]", title = "????", publisher = "Gauthier-Villars", address = "Paris, France", pages = "253--312", year = "1868", bibdate = "Tue Mar 14 18:45:06 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, author-dates = "1736--1813", language = "French", remark = "See especially pages 267 and 272.", } @Article{Schlesinger:1911:GJA, author = "L. Schlesinger", title = "{{\"U}ber Gauss' Jugendarbeiten zum arithmetisch--geometrischen Mittel}. ({German}) [{On} {Gauss}' youthful work on the arithmetic--geometric mean]", journal = j-DEUTSCH-MATH-V, volume = "20", number = "??", pages = "396--403", month = "????", year = "1911", bibdate = "Tue Mar 14 18:38:44 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", ZMnumber = "42.0466.01", acknowledgement = ack-nhfb, ajournal = "Jber. Deutsch. Math.--Verein", fjournal = "Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)", language = "German", } @InCollection{Gauss:1917:OPG, author = "Carl Friedrich Gauss", booktitle = "Werke", title = "De origene propietatibusque generalibus numerorum mediorum aritmet. geometricorum. ({Latin}) []", volume = "X-1", publisher = "Koniglichen Gesellschaft der Wissenschaften", address = "G{\"o}ttingen, Germany", pages = "??--??", year = "1917", bibdate = "Tue Mar 14 17:28:06 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, author-dates = "1777--1855", language = "Latin", remark = "This work was not published during Gauss's lifetime (1777--1855).", } @Article{King:1921:SNF, author = "Louis Vessot King", title = "On Some New Formulae for the Numerical Calculation of the Mutual Induction of Coaxial Circles", journal = j-PROC-R-SOC-LOND-SER-A-MATH-PHYS, volume = "100", number = "702", pages = "60--66", day = "4", month = oct, year = "1921", DOI = "https://doi.org/10.1098/rspa.1921.0070", ISSN = "0950-1207 (print), 2053-9150 (electronic)", bibdate = "Wed Feb 03 09:07:10 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", note = "This is the first known publication of the AGM method, discovered by the author in 1913, for computing Jacobian elliptic functions. See also \cite{King:1924:DNC,King:2007:DNC}.", URL = "http://www.jstor.org/stable/93861", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character", journal-URL = "http://rspa.royalsocietypublishing.org/", } @Book{King:1924:DNC, author = "Louis Vessot King", title = "On the Direct Numerical Calculation of Elliptic Functions and Integrals", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "viii + 42", year = "1924", LCCN = "QA343", bibdate = "Wed Feb 03 08:53:04 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", acknowledgement = ack-nhfb, remark = "The AGM method for Jacobian elliptic functions was discovered by this book's author at McGill University in 1913, first published in \cite{King:1921:SNF}, and then in this monograph (reprinted in \cite{King:2007:DNC}).", } @Article{Huntington:1927:SIP, author = "Edward V. Huntington", title = "Sets of independent postulates for the arithmetic mean, the geometric mean, the harmonic mean, and the root-mean-square", journal = j-TRANS-AM-MATH-SOC, volume = "29", number = "1", pages = "1--22", year = "1927", CODEN = "TAMTAM", DOI = "https://doi.org/10.2307/1989276", ISSN = "0002-9947 (print), 1088-6850 (electronic)", ISSN-L = "0002-9947", MRclass = "26D15", MRnumber = "1501374", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Transactions of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/tran/", } @Article{Geppert:1928:TAG, author = "Harald Geppert", title = "{Zur Theorie des arithmetisch--geometrischen Mittels}. ({German}) [{On} the theory of the arithmetic--geometric mean]", journal = j-MATH-ANN, volume = "99", number = "1", pages = "162--180", month = dec, year = "1928", CODEN = "MAANA3", DOI = "https://doi.org/10.1007/BF01459092", ISSN = "0025-5831 (print), 1432-1807 (electronic)", ISSN-L = "0025-5831", bibdate = "Tue Mar 14 18:33:51 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/article/10.1007/BF01459092", acknowledgement = ack-nhfb, fjournal = "Mathematische Annalen", journal-URL = "http://link.springer.com/journal/208", language = "German", received = "04 March 1927", } @PhdThesis{Butter:1936:CTA, author = "Franklin A. {Butter, Jr.}", title = "A Contribution to the Theory of the Arithmetic--Geometric Mean", type = "Thesis ({Ph.D.})", school = "Stanford University", address = "Stanford, CA, USA", pages = "????", year = "1936", MRnumber = "2937121", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://search.proquest.com/docview/301788817", acknowledgement = ack-nhfb, } @PhdThesis{Rauch:1942:MPS, author = "Stanley Eugene Rauch", title = "Mapping properties of the second arithmetic mean of the geometric series", type = "Thesis ({Ph.D.})", school = "Stanford University", address = "Stanford, CA, USA", pages = "????", year = "1942", MRnumber = "2937549", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://search.proquest.com/docview/301867689", acknowledgement = ack-nhfb, } @Article{Stubban:1944:AGM, author = "John Olav Stubban", title = "On the arithmetic and geometric means", journal = "Norsk Mat. Tidsskr.", volume = "26", pages = "116--117", year = "1944", MRclass = "27.0X", MRnumber = "0017777", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Aiyer:1945:AGM, author = "S. Janardana Aiyer", title = "On the arithmetic and the geometric means from a type {III} population", journal = j-MATH-STUDENT, volume = "13", pages = "11--15", year = "1945", CODEN = "MTHSBH", ISSN = "0025-5742", MRclass = "62.0X", MRnumber = "0013881", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematics Student", } @Article{Nanjundiah:1946:IRA, author = "T. S. Nanjundiah", title = "Inequalities relating to arithmetic and geometric means. {I}, {II}", journal = "Half-Yearly J. Mysore Univ. Sect. B., N.S.", volume = "6", pages = "63--77, 107--113", year = "1946", MRclass = "27.0X", MRnumber = "0044588", MRreviewer = "E. F. Beckenbach", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Bellman:1956:AGM, author = "Richard Bellman", title = "On the arithmetic--geometric mean inequality", journal = j-MATH-STUDENT, volume = "24", pages = "233--234 (1957)", year = "1956", CODEN = "MTHSBH", ISSN = "0025-5742", MRclass = "09.2X", MRnumber = "0085208", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematics Student", zzbibdate = "Tue Aug 15 11:29:11 2017", } @Article{Hunter:1956:GIA, author = "John Hunter", title = "A generalization of the inequality of the arithmetic--geometric means", journal = "Proc. Glasgow Math. Assoc.", volume = "2", pages = "149--158", year = "1956", MRclass = "10.2X", MRnumber = "0075984", MRreviewer = "Harvey Cohn", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, zzbibdate = "Tue Aug 15 11:29:11 2017", } @Article{Torrent:1956:NEC, author = "J. Maj{\'o} Torrent", title = "Note on the extension to the complex field of the arithmetic and the geometric mean", journal = "Gac. Mat., Madrid (1)", volume = "8", pages = "195--198", year = "1956", MRclass = "09.2X", MRnumber = "0085209", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Kober:1958:AGM, author = "H. Kober", title = "On the arithmetic and geometric means and on {H{\"o}lder}'s inequality", journal = j-PROC-AM-MATH-SOC, volume = "9", pages = "452--459", year = "1958", CODEN = "PAMYAR", DOI = "https://doi.org/10.2307/2033003", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "26.00", MRnumber = "0093564", MRreviewer = "T. M. Apostol", bibdate = "Tue Aug 15 11:29:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Diananda:1960:CNS, author = "P. H. Diananda", title = "Classroom Notes: a Simple Proof of the Arithmetic Mean Geometric Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "67", number = "10", pages = "1007--1007", month = dec, year = "1960", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2309236", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRnumber = "1531004", bibdate = "Mon Jun 28 12:36:07 MDT 1999", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Akerberg:1963:CNP, author = "Bengt Akerberg", title = "Classroom Notes: A Proof of the Arithmetic--Geometric Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "70", number = "9", pages = "997--998", month = nov, year = "1963", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2313068", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "Contributed Item", MRnumber = "1532378", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Everitt:1963:IGA, author = "W. N. Everitt", title = "On an Inequality for the Generalized Arithmetic and Geometric Means", journal = j-AMER-MATH-MONTHLY, volume = "70", number = "3", pages = "251--255", month = mar, year = "1963", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Jun 28 12:37:02 MDT 1999", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Wilf:1963:SAI, author = "Herbert S. Wilf", title = "Some applications of the inequality of arithmetic and geometric means to polynomial equations", journal = j-PROC-AM-MATH-SOC, volume = "14", pages = "263--265", year = "1963", CODEN = "PAMYAR", DOI = "https://doi.org/10.2307/2034624", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "30.10", MRnumber = "0145047", MRreviewer = "O. Shisha", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Oppenheim:1965:ICA, author = "Alexander Oppenheim", title = "On inequalities connecting arithmetic means and geometric means of two sets of three positive numbers", journal = j-MATH-GAZ, volume = "49", pages = "160--162", year = "1965", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3612307", ISSN = "0025-5572 (print), 2056-6328 (electronic)", MRclass = "26.70", MRnumber = "0185063", MRreviewer = "H. Burkill", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Book{Pars:1965:TAD, author = "Leopold Alexander Pars", title = "A Treatise on Analytical Dynamics", publisher = "Heinemann", address = "London, UK", pages = "xxi + 641", year = "1965", bibdate = "Wed Mar 15 08:09:52 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Tricomi:1965:SID, author = "F. G. Tricomi", title = "Sull'algoritmo iterativo del {Borchardt} e su di una sua generalizzazione. ({Italian}) [{On} the iterative algorithm of {Borchardt} and on one of its generalization]", journal = j-REND-CIRC-MAT, volume = "2", number = "14", pages = "85--94", month = "????", year = "1965", CODEN = "RCMMAR", ISSN = "0009-725X (print), 1973-4409 (electronic)", ISSN-L = "0009-725X", bibdate = "Tue Mar 14 18:46:58 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Rendiconti del Circolo matematico di Palermo", } @Article{Mitrinovic:1966:ICA, author = "D. S. Mitrinovi{\'c}", title = "An inequality concerning the arithmetic and geometric means", journal = j-MATH-GAZ, volume = "50", pages = "310--311", year = "1966", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3614693", ISSN = "0025-5572 (print), 2056-6328 (electronic)", MRclass = "26.70", MRnumber = "0229768", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Article{Shisha:1966:GII, author = "O. Shisha", title = "Geometrical interpretations of the inequalities between the arithmetic, geometric and harmonic means", journal = j-MATH-MAG, volume = "39", pages = "268--269", year = "1966", CODEN = "MAMGA8", DOI = "https://doi.org/10.2307/2689010", ISSN = "0011-801x", MRclass = "26.70", MRnumber = "0202951", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Delta. University of Wisconsin", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @TechReport{Tricomi:1966:RUS, author = "F. G. Tricomi", title = "Lectures on the use of special functions by calculations with electronic computers", type = "Lecture Series", number = "47", institution = "The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park", address = "College Park, MD, USA", year = "1966", bibdate = "Tue Mar 14 18:48:58 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Bullen:1967:SMI, author = "P. S. Bullen", title = "Some more inequalities involving the arithmetic and geometric means", journal = "Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No.", volume = "181--196", pages = "61--66", year = "1967", ISSN = "0522-8441", MRclass = "26.70", MRnumber = "0223515", MRreviewer = "Z. Dar{\'o}czy", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Univerzitet u Beogradu. Publikacije Elektrotehni\v ckog Fakulteta. Serija Matematika i Fizika", } @Article{Everitt:1967:LPA, author = "W. N. Everitt", title = "On a limit problem associated with the arithmetic--geometric mean inequality", journal = j-J-LOND-MATH-SOC, volume = "42", pages = "712--718", year = "1967", CODEN = "JLMSAK", DOI = "https://doi.org/10.1112/jlms/s1-42.1.712", ISSN = "0024-6107 (print), 1469-7750 (electronic)", ISSN-L = "0024-6107", MRclass = "26.70", MRnumber = "0222229", MRreviewer = "P. H. Diananda", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "See corrigendum \cite{Everitt:1969:CLP}.", acknowledgement = ack-nhfb, fjournal = "Journal of the London Mathematical Society. Second Series", journal-URL = "http://jlms.oxfordjournals.org/content/by/year", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Gaines:1967:AMG, author = "Fergus Gaines", title = "Classroom Notes: On the Arithmetic Mean--Geometric Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "74", number = "3", pages = "305--306", month = mar, year = "1967", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2316036", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26.70 (15.00)", MRnumber = "0214715", MRreviewer = "A. Jaeger", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Klamkin:1968:ICA, author = "Murray S. Klamkin", title = "Inequalities concerning the arithmetic, geometric and harmonic means", journal = j-MATH-GAZ, volume = "52", pages = "156--157", year = "1968", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3612687", ISSN = "0025-5572 (print), 2056-6328 (electronic)", MRclass = "26.70", MRnumber = "0229769", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Article{Oppenheim:1968:ICA, author = "A. Oppenheim", title = "On inequalities connecting arithmetic means and geometric means of two sets of three positive numbers. {II}", journal = "Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No.", volume = "210-228", pages = "21--24", year = "1968", ISSN = "0522-8441", MRclass = "26.70", MRnumber = "0231960", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Univerzitet u Beogradu. Publikacije Elektrotehni\v ckog Fakulteta. Serija Matematika i Fizika", } @Article{OShea:1968:MNA, author = "Siobhan O'Shea", title = "Mathematical Notes: The Arithmetic Geometric Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "75", number = "10", pages = "1092--1093", month = dec, year = "1968", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2315738", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "Contributed Item", MRnumber = "1535171", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Book{Pars:1968:TAD, author = "Leopold Alexander Pars", title = "A Treatise on Analytical Dynamics", publisher = "Heinemann", address = "London, UK", pages = "xxi + 641", year = "1968", ISBN = "0-435-52690-1", ISBN-13 = "978-0-435-52690-0", bibdate = "Wed Mar 15 08:09:52 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, remark = "Reprint of \cite{Pars:1965:TAD} with corrections.", } @Article{Everitt:1969:CLP, author = "W. N. Everitt", title = "Corrigendum: {``On a limit problem associated with the arithmetic--geometric mean inequality''}", journal = j-J-LOND-MATH-SOC-2, volume = "1", pages = "428--430", year = "1969", CODEN = "JLMSAK", DOI = "https://doi.org/10.1112/jlms/s2-1.1.428-s", ISSN = "0024-6107 (print), 1469-7750 (electronic)", ISSN-L = "0024-6107", MRclass = "26.70", MRnumber = "0248309", MRreviewer = "P. H. Diananda", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "See \cite{Everitt:1967:LPA}.", acknowledgement = ack-nhfb, fjournal = "Journal of the London Mathematical Society. Second Series", journal-URL = "http://jlms.oxfordjournals.org/content/by/year", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Carlson:1970:IMA, author = "B. C. Carlson", title = "An Inequality of Mixed Arithmetic and Geometric Means", journal = j-SIAM-REVIEW, volume = "12", number = "2", pages = "287--288", month = "????", year = "1970", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1012054", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:06:17 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/12/2; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "April 1970", } @Article{Lehmer:1970:CCM, author = "D. H. Lehmer", title = "On the compounding of certain means", journal = j-NAMS, volume = "17", number = "??", pages = "634--635", month = "????", year = "1970", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", bibdate = "Tue Mar 14 19:09:32 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", remark = "Check title: not found in MathSciNet or zbMATH databases, or in Web searches, but cited in \cite[ref. 17]{Carlson:1971:AIA}.", } @Article{Loewner:1970:DBG, author = "Charles Loewner and Henry B. Mann", title = "On the difference between the geometric and the arithmetic mean of {$n$} quantities", journal = j-ADV-MATH, volume = "5", pages = "472--473 (1970)", year = "1970", CODEN = "ADMTA4", DOI = "https://doi.org/10.1016/0001-8708(70)90012-5", ISSN = "0001-8708 (print), 1090-2082 (electronic)", ISSN-L = "0001-8708", MRclass = "26.70", MRnumber = "0279259", MRreviewer = "P. H. Diananda", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Advances in Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/00018708", } @Article{Mitrovic:1970:SII, author = "{\v{Z}}arko Mitrovi{\'c}", title = "Some inequalities involving elementary symmetric function and arithmetic and geometric means", journal = j-MATH-GAZ, volume = "54", pages = "155--157", year = "1970", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3612110", ISSN = "0025-5572 (print), 2056-6328 (electronic)", MRclass = "26.70", MRnumber = "0264010", MRreviewer = "V. Ganapathy Iyer", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Article{Carlson:1971:AIA, author = "Bille Chandler Carlson", title = "Algorithms Involving Arithmetic and Geometric Means", journal = j-AMER-MATH-MONTHLY, volume = "78", number = "5", pages = "496--505", month = may, year = "1971", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2317754", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "33.19", MRnumber = "0283246", MRreviewer = "F. G{\"o}tze", bibdate = "Mon Jun 28 12:36:21 MDT 1999", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; JSTOR database", URL = "http://www.jstor.org/stable/2317754", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Carlson:1971:MAG, author = "B. C. Carlson and R. K. Meany and S. A. Nelson", title = "Mixed arithmetic and geometric means", journal = j-PAC-J-MATH, volume = "38", pages = "343--349", year = "1971", CODEN = "PJMAAI", ISSN = "0030-8730 (print), 1945-5844 (electronic)", ISSN-L = "0030-8730", MRclass = "26A87", MRnumber = "0304590", MRreviewer = "H. Kober", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://projecteuclid.org/euclid.pjm/1102970046", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Mathematics", journal-URL = "http://msp.org/pjm", } @Article{Lehmer:1971:CCM, author = "D. H. Lehmer", title = "On the compounding of certain means", journal = j-J-MATH-ANAL-APPL, volume = "36", number = "1", pages = "183--200", month = oct, year = "1971", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/0022-247X(71)90029-1", ISSN = "0022-247X (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "10.43", MRnumber = "281696", MRreviewer = "D. Rearick", bibdate = "Tue Mar 14 18:52:11 2017", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/l/lehmer-derrick-henry.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://www.sciencedirect.com/science/article/pii/0022247X71900291", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "arithmetic--geometric mean (AGM) iteration; complete elliptic integrals of the first and second kinds. Landen s transformation", } @Article{Meany:1971:BRB, author = "R. K. Meany and S. A. Nelson and B. C. Carlson", title = "Book Review: {{\booktitle{An Inequality of Mixed Arithmetic and Geometric Means}} (B. C. Carlson)}", journal = j-SIAM-REVIEW, volume = "13", number = "2", pages = "253--255", month = "????", year = "1971", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1013058", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Thu Mar 27 09:06:28 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/13/2; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "April 1971", } @Article{Morita:1971:CLG, author = "Tohru Morita and Tsuyoshi Horiguchi", title = "Calculation of the lattice {Green}'s function for the bcc, fcc, and rectangular lattices", journal = j-J-MATH-PHYS, volume = "12", number = "6", pages = "986--992", month = jun, year = "1971", CODEN = "JMAPAQ", DOI = "https://doi.org/10.1063/1.1665693", ISSN = "0022-2488 (print), 1089-7658 (electronic), 1527-2427", ISSN-L = "0022-2488", bibdate = "Fri Oct 28 16:39:38 MDT 2011", bibsource = "http://jmp.aip.org/; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys1970.bib", URL = "http://jmp.aip.org/resource/1/jmapaq/v12/i6/p986_s1", acknowledgement = ack-nhfb, classification = "A0550 (Lattice theory and statistics; Ising problems)", corpsource = "Tohoku Univ., Sendai, Japan", fjournal = "Journal of Mathematical Physics", journal-URL = "http://jmp.aip.org/", keywords = "arithmetic mean; bcc lattice; complex modulus; divergence; elliptic integral; fcc lattice; geometric mean; Green's function methods; Lattice Green function; lattice theory and statistics; rectangular lattice", onlinedate = "28 October 2003", pagecount = "7", treatment = "T Theoretical or Mathematical", } @Article{Passy:1971:GWM, author = "U. Passy", title = "Generalized Weighted Mean Programming", journal = j-SIAM-J-APPL-MATH, volume = "20", number = "4", pages = "763--778", month = jun, year = "1971", CODEN = "SMJMAP", ISSN = "0036-1399 (print), 1095-712X (electronic)", ISSN-L = "0036-1399", bibdate = "Thu Oct 15 18:16:06 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamjapplmath.bib; JSTOR database", acknowledgement = ack-nhfb, classification = "B0260 (Optimisation techniques); C1180 (Optimisation techniques)", corpsource = "Tech. Israel Inst. Technol., Haifa, Israel", fjournal = "SIAM Journal on Applied Mathematics", journal-URL = "http://epubs.siam.org/siap", keywords = "algorithms; arithmetic geometric mean inequality; nonlinear programming; weighted mean programming", treatment = "T Theoretical or Mathematical", } @Article{Hering:1972:GAG, author = "Franz Hering", title = "A generalization of the arithmetic--geometric mean inequality and an application to finite sequences of zeros and ones", journal = j-ISRAEL-J-MATH, volume = "11", number = "1", pages = "14--30", year = "1972", CODEN = "ISJMAP", DOI = "https://doi.org/10.1007/BF02761445", ISSN = "0021-2172 (print), 1565-8511 (electronic)", ISSN-L = "0021-2172", MRclass = "05A17", MRnumber = "0302466", MRreviewer = "S. G. Williamson", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/article/10.1007/BF02761445", acknowledgement = ack-nhfb, fjournal = "Israel Journal of Mathematics", journal-URL = "http://link.springer.com/journal/11856", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Morita:1972:CAG, author = "Tohru Morita and Tsuyoshi Horiguchi", title = "Convergence of the arithmetic--geometric mean procedure for the complex variables and the calculation of the complete elliptic integrals with complex modulus", journal = j-NUM-MATH, volume = "20", number = "5", pages = "425--430", month = oct, year = "1972", CODEN = "NUMMA7", DOI = "https://doi.org/10.1007/BF01402565", ISSN = "0029-599X (print), 0945-3245 (electronic)", ISSN-L = "0029-599X", MRclass = "65D20", MRnumber = "0315870", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/nummath.bib", URL = "http://www.springerlink.com/openurl.asp?genre=article&issn=0029-599X&volume=20&issue=5&spage=425", abstract = "The convergence of the arithmetic--geometric mean procedure is checked for complex variables. The procedure is shown to be useful for the evaluation of the complete elliptic integrals of the first and second kinds with complex modulus. It is suggested that the procedure will be useful also for the numerical calculation of the elliptic integrals and the Jacobian elliptic functions with complex modulus in general.", acknowledgement = ack-nhfb, fjournal = "Numerische Mathematik", journal-URL = "http://link.springer.com/journal/211", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Thurston:1972:HGU, author = "H. A. Thurston", title = "How good is the usual approximation for the period of a simple pendulum?", journal = j-MATH-GAZ, volume = "56", number = "??", pages = "120--122", month = "????", year = "1972", CODEN = "MAGAAS", ISSN = "0025-5572 (print), 2056-6328 (electronic)", bibdate = "Wed Mar 15 07:24:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", keywords = "arithmetic--geometric mean", } @Article{Vasic:1972:ICA, author = "Petar M. Vasi{\'c}", title = "On inequalities connecting arithmetic means and geometric means of two sets of $n$ positive numbers", journal = "Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.", volume = "381--409", pages = "63--66", year = "1972", ISSN = "0522-8441", MRclass = "26A86", MRnumber = "0333098", MRreviewer = "Victor I. Levin", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Univerzitet u Beogradu. Publikacije Elektrotehni\v ckog Fakulteta. Serija Matematika i Fizika", } @Article{Tung:1975:LUB, author = "S. H. Tung", title = "On lower and upper bounds of the difference between the arithmetic and the geometric mean", journal = j-MATH-COMPUT, volume = "29", number = "131", pages = "834--836", month = jul, year = "1975", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2005294", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "26A87", MRnumber = "0393393", MRreviewer = "D. C. Benson", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0240 (Probability and statistics); C1140 (Probability and statistics)", corpsource = "Dept. of Math. and Statistics, Miami Univ., Oxford, OH, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "arithmetic mean; geometric mean; lower bounds; statistics; upper bounds", treatment = "T Theoretical or Mathematical", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Brent:1976:FMP, author = "Richard P. Brent", title = "Fast Multiple-Precision Evaluation of Elementary Functions", journal = j-J-ACM, volume = "23", number = "2", pages = "242--251", month = apr, year = "1976", CODEN = "JACOAH", DOI = "https://doi.org/10.1145/321941.321944", ISSN = "0004-5411 (print), 1557-735X (electronic)", ISSN-L = "0004-5411", MRclass = "68A20 (68A10)", MRnumber = "52 \#16111", MRreviewer = "Amnon Barak", bibdate = "Wed Jan 15 18:12:53 MST 1997", bibsource = "Compendex database; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jacm.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", abstract = "Let $ f(x) $ be one of the usual elementary functions ($ \exp $, $ \log $, $ \arctan $, $ \sin $, $ \cosh $, etc.), and let $ M(n) $ be the number of single-precision operations required to multiply $n$-bit integers. It is shown that $ f(x) $ can be evaluated, with relative error $ O(2 - n) $, in $ O(M(n)l o g (n)) $ operations as $ n \rightarrow \infty $, for any floating-point number $x$ (with an $n$-bit fraction) in a suitable finite interval. From the Sch{\"o}nhage--Strassen bound on $ M(n) $, it follows that an $n$-bit approximation to $ f(x) $ may be evaluated in $ O(n \log_(n) \log \log (n)) $ operations. Special cases include the evaluation of constants such as $ \pi^e$, and $ e^\pi $. The algorithms depend on the theory of elliptic integrals, using the arithmetic--geometric mean iteration and ascending Landen transformations.", acknowledgement = ack-nhfb, classification = "723", fjournal = "Journal of the Association for Computing Machinery", journal-URL = "http://portal.acm.org/browse_dl.cfm?idx=J401", journalabr = "J Assoc Comput Mach", keywords = "computational complexity; computer arithmetic; computer programming", } @InProceedings{Brent:1976:MPZ, author = "Richard P. Brent", title = "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", crossref = "Traub:1976:ACC", pages = "151--176", year = "1976", MRclass = "68A20", MRnumber = "54 \#11843", MRreviewer = "Claus-Peter Schnorr", bibdate = "Sat Jan 11 17:44:01 MST 1997", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, } @Article{Chong:1976:AMG, author = "Kong Ming Chong", title = "The arithmetic mean-geometric mean inequality: a new proof", journal = j-MATH-MAG, volume = "49", number = "2", pages = "87--88", year = "1976", CODEN = "MAMGA8", DOI = "https://doi.org/10.2307/2689438", ISSN = "0011-801x", MRclass = "26A86", MRnumber = "0399388", MRreviewer = "Peter Bullen", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Delta. University of Wisconsin", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Fink:1976:GAG, author = "A. M. Fink and Max {Jodeit, Jr.}", title = "A generalization of the arithmetic--geometric means inequality", journal = j-PROC-AM-MATH-SOC, volume = "61", number = "2", pages = "255--261 (1977)", year = "1976", CODEN = "PAMYAR", DOI = "https://doi.org/10.2307/2041321", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "26A86", MRnumber = "0427564", MRreviewer = "E. F. Beckenbach", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Glaser:1976:RGM, author = "Ronald E. Glaser", title = "The ratio of the geometric mean to the arithmetic mean for a random sample from a gamma distribution", journal = j-J-AM-STAT-ASSOC, volume = "71", number = "354", pages = "480--487", year = "1976", CODEN = "JSTNAL", ISSN = "0162-1459 (print), 1537-274x (electronic)", ISSN-L = "0162-1459", MRclass = "62E15", MRnumber = "0403011", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://links.jstor.org/sici?sici=0162-1459(197606)71:354<480:TROTGM>2.0.CO;2-I&origin=MSN", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Salamin:1976:CUA, author = "Eugene Salamin", title = "Computation of $ \pi $ Using Arithmetic--Geometric Mean", journal = j-MATH-COMPUT, volume = "30", number = "135", pages = "565--570", month = jul, year = "1976", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2005327", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "10A30 (10A40 33A25)", MRnumber = "0404124 (53 \#7928)", MRreviewer = "I. John Zucker", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1970.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database; MathSciNet database", note = "See also \cite{Brent:1976:MPZ,Brent:2010:MPZ}.", URL = "http://www.jstor.org/stable/2005327", ZMnumber = "0345.10003", acknowledgement = ack-nhfb, citedby = "Fullerton:1980:BEM", classcodes = "B0290D (Functional analysis); C4120 (Functional analysis)", corpsource = "Charles Stark Draper Lab., Cambridge, MA, USA", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "arithmetic geometric mean; convergence; elliptic integrals; error analysis; fast Fourier transform multiplication; function evaluation; Landen's; Legendre's relation; numerical computation of pi; transformation", remark = "Fullerton: A quadratically convergent algorithm.", treatment = "A Application; T Theoretical or Mathematical", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Chong:1977:AMG, author = "Kong Ming Chong", title = "On the arithmetic-mean-geometric-mean inequality", journal = "Nanta Math.", volume = "10", number = "1", pages = "26--27", year = "1977", ISSN = "0077-2739", MRclass = "26A86", MRnumber = "0460568", MRreviewer = "G. Sansone", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Nanyang University. Nanta Mathematica", } @Article{Schaumberger:1977:APA, author = "Norman Schaumberger and Bert Kabak", title = "Another proof of the arithmetic--geometric mean inequality", journal = j-PI-MU-EPSILON-J, volume = "6", number = "6", pages = "352--354", year = "1977", CODEN = "PMEJBR", ISSN = "0031-952x", MRclass = "26A86", MRnumber = "0430190", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Pi Mu Epsilon Journal", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Almkvist:1978:AGM, author = "Gert Almkvist", title = "Aritmetisk--geometriska Medelv{\"a}rdet och Ellipsens B{\aa}gl{\"a}ngd. ({Swedish}) [{The} arithmetic--geometric mean and the arc length of the ellipse]", journal = j-NORDISK-MATH-TIDSKR, volume = "25--26", number = "3--4", pages = "121--130, 208", year = "1978", ISSN = "0029-1412, 0801-3500", MRclass = "10D05 (33A25)", MRnumber = "561786", MRreviewer = "Troels J{\o}rgensen", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/24525291", acknowledgement = ack-nhfb, fjournal = "Nordisk Matematisk Tidskrift", journal-URL = "http://www.jstor.org/journal/nordmatetids", language = "Swedish", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Bajpai:1978:SAG, author = "S. K. Bajpai", title = "Special arithmetic and geometric means preserve {$ \Phi $}-like univalence", journal = "Rev. Colombiana Mat.", volume = "12", number = "3-4", pages = "83--90", year = "1978", ISSN = "0034-7426", MRclass = "30C45", MRnumber = "533713", MRreviewer = "P. T. Mocanu", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Revista Colombiana de Matem{\'a}ticas", } @Article{Cartwright:1978:RAM, author = "D. I. Cartwright and M. J. Field", title = "A refinement of the arithmetic mean-geometric mean inequality", journal = j-PROC-AM-MATH-SOC, volume = "71", number = "1", pages = "36--38", year = "1978", CODEN = "PAMYAR", DOI = "https://doi.org/10.2307/2042211", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "26A87", MRnumber = "0476971", MRreviewer = "V. Ganapathy Iyer", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Landsberg:1978:TPI, author = "P. T. Landsberg", title = "A thermodynamic proof of the inequality between arithmetic and geometric mean", journal = j-PHYS-LET-A, volume = "67", number = "1", pages = "1", year = "1978", CODEN = "PYLAAG", DOI = "https://doi.org/10.1016/0375-9601(78)90548-0", ISSN = "0031-9163 (print), 1873-2410 (electronic)", ISSN-L = "0375-9601", MRclass = "80A10 (26D20)", MRnumber = "601317", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Physics Letters. A", journal-URL = "http://www.sciencedirect.com/science/journal/03759601", } @Article{Schaumberger:1978:CPA, author = "Norman Schaumberger", title = "A Calculus Proof of the Arithmetic-Geometric Mean Inequality", journal = j-TWO-YEAR-COLL-MATH-J, volume = "9", number = "1", pages = "16--17", month = jan, year = "1978", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1978.11974538", ISSN = "0049-4925 (print), 2325-9116 (electronic)", ISSN-L = "0049-4925", bibdate = "Thu Feb 14 09:48:43 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00494925.1978.11974538", acknowledgement = ack-nhfb, fjournal = "Two-Year College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; http://www.jstor.org/journals/00494925.html", onlinedate = "30 Jan 2018", } @Article{Abriata:1979:CTP, author = "J. P. Abriata", title = "Comment on a thermodynamic proof of the inequality between arithmetic and geometric mean", journal = j-PHYS-LET-A, volume = "71", number = "4", pages = "309--310", year = "1979", CODEN = "PYLAAG", DOI = "https://doi.org/10.1016/0375-9601(79)90061-6", ISSN = "0031-9163 (print), 1873-2410 (electronic)", ISSN-L = "0375-9601", MRclass = "80A10 (26D20)", MRnumber = "588726", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Physics Letters. A", journal-URL = "http://www.sciencedirect.com/science/journal/03759601", } @InCollection{Krafft:1979:RGA, author = "Olaf Krafft and Rudolf Mathar and Martin Schaefer", booktitle = "Numerische {Methoden} bei graphentheoretischen und kombinatorischen {Problemen}, {Band} 2 ({Tagung}, {Math}. {Forschungsinst}., {Oberwolfach}, 1978)", title = "A refined geometric--arithmetic means inequality for integers", volume = "46", publisher = "Birkh{\"a}user, Basel-Boston, Mass.", pages = "216--223", year = "1979", MRclass = "26D20 (10E99)", MRnumber = "562279", MRreviewer = "Norbert Schmitz", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Internat. Ser. Numer. Math.", acknowledgement = ack-nhfb, } @Book{Pars:1979:TAD, author = "Leopold Alexander Pars", title = "A Treatise on Analytical Dynamics", publisher = "Ox Bow Press", address = "Woodbridge, CT, USA", pages = "xxi + 641", year = "1979", ISBN = "0-918024-07-2", ISBN-13 = "978-0-918024-07-7", LCCN = "QA845 .P32 1979", bibdate = "Wed Mar 15 08:12:22 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, remark = "Reprint. Originally published: New York: Wiley, 1965.", subject = "Dynamics", tableofcontents = "Motion of a particle \\ Dynamical Systems \\ The first form of the fundamental equation \\ The second and third forms of the fundamental equation \\ Lagrangian coordinates \\ Lagrange's equations \\ The theory of rotations \\ First applications of Lagrange's equations \\ The theory of vibrations \\ Further applications of Lagrange's equations \\ Variable mass \\ The Gibbs--Appell equations \\ Applications of the Gibbs--Appell equations \\ Impulsive motion \\ The sixth form of the fundamental equation \\ The Hamilton--Jacobi theorem \\ Separable systems with n degrees of freedom \\ Systems with one degree of freedom, motion near a singular point \\ Systems with one degree of freedom, the cyclic characteristics \\ Systems with n degrees of freedom, properties of the characteristics \\ Hamilton's equations \\ Motion in the neighborhood of a given motion, stability in motion \\ Contact transformations \\ Transformation theory \\ Variation principles \\ The principle of Least Action \\ The restricted problem of three bodies \\ The problem of three bodies \\ Periodic orbits", } @Article{Nandi:1980:EDN, author = "S. B. Nandi", title = "On the exact distribution of a normalized ratio of the weighted geometric mean to the unweighted arithmetic mean in samples from gamma distributions", journal = j-J-AM-STAT-ASSOC, volume = "75", number = "369", pages = "217--220", year = "1980", CODEN = "JSTNAL", ISSN = "0003-1291", ISSN-L = "0162-1459", MRclass = "62E15 (62F05)", MRnumber = "568595", MRreviewer = "G. S. Lingappaiah", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://links.jstor.org/sici?sici=0162-1459(198003)75:369<217:OTEDOA>2.0.CO;2-M&origin=MSN", acknowledgement = ack-nhfb, fjournal = "Journal of the American Statistical Association", journal-URL = "http://www.tandfonline.com/loi/uasa20", } @Article{Cusmariu:1981:MNP, author = "Adolf Cusmariu", title = "Mathematical Notes: a Proof of the Arithmetic Mean--Geometric Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "88", number = "3", pages = "192--194", month = mar, year = "1981", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2320467", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26D15 (05A20 40A99)", MRnumber = "619566 (82g:26024)", MRreviewer = "D. C. Russell", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1980.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Fink:1981:WAG, author = "A. M. Fink", title = "A weighted-arithmetic--geometric means inequality", journal = "Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.", volume = "716--734", number = "716-734", pages = "35--40", year = "1981", ISSN = "0522-8441", MRclass = "26D15", MRnumber = "642007", MRreviewer = "Peter Bullen", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Univerzitet u Beogradu. Publikacije Elektrotehni\v ckog Fakulteta. Serija Matematika i Fizika", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Heinrich:1981:VAG, author = "H. Heinrich", title = "{Eine Verallgemeinerung des arithmetisch--geometrischen Mittels}. ({German}) [{A} generalization of the arithmetic--geometric mean]", journal = j-Z-ANGE-MATH-MECH, volume = "61", number = "6", pages = "265--267", year = "1981", CODEN = "ZAMMAX", DOI = "https://doi.org/10.1002/zamm.19810610610", ISSN = "0044-2267 (print), 1521-4001 (electronic)", ISSN-L = "0044-2267", MRclass = "26D15", MRnumber = "638023", MRreviewer = "E. F. Beckenbach", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik. Ingenieurwissenschaftliche Forschungsarbeiten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-4001", language = "German", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Perisastry:1982:BRA, author = "M. Perisastry and V. N. Murty", title = "Bounds for the Ratio of the Arithmetic Mean to the Geometric Mean", journal = j-TWO-YEAR-COLL-MATH-J, volume = "13", number = "2", pages = "160--161", month = mar, year = "1982", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1982.11972600", ISSN = "0049-4925 (print), 2325-9116 (electronic)", ISSN-L = "0049-4925", bibdate = "Thu Feb 14 09:49:31 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00494925.1982.11972600", acknowledgement = ack-nhfb, fjournal = "Two-Year College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; http://www.jstor.org/journals/00494925.html", onlinedate = "30 Jan 2018", } @Article{Schaumberger:1982:SAP, author = "Norman Schaumberger", title = "Still Another Proof of the Arithmetic-Geometric Mean Inequality", journal = j-TWO-YEAR-COLL-MATH-J, volume = "13", number = "2", pages = "159--160", month = mar, year = "1982", CODEN = "????", DOI = "https://doi.org/10.1080/00494925.1982.11972599", ISSN = "0049-4925 (print), 2325-9116 (electronic)", ISSN-L = "0049-4925", bibdate = "Thu Feb 14 09:49:31 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/00494925.1982.11972599", acknowledgement = ack-nhfb, fjournal = "Two-Year College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; http://www.jstor.org/journals/00494925.html", onlinedate = "30 Jan 2018", } @Article{Ando:1983:AGH, author = "T. Ando", title = "On the arithmetic--geometric-harmonic-mean inequalities for positive definite matrices", journal = j-LINEAR-ALGEBRA-APPL, volume = "52/53", pages = "31--37", year = "1983", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(83)80005-6", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45 (15A48 47A60)", MRnumber = "709342", MRreviewer = "George P. Barker", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @Article{Borwein:1983:GAG, author = "D. Borwein and P. B. Borwein", title = "A Generalized Arithmetic--Geometric Mean", journal = j-SIAM-REVIEW, volume = "25", number = "3", pages = "401--401", month = "????", year = "1983", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1025081", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:53:39 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/25/3; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "July 1983", } @Article{Ono:1983:GGT, author = "Takashi Ono", title = "A generalization of {Gauss}' theorem on arithmetic--geometric means", journal = j-PROC-JAPAN-ACAD-SER-A-MATH-SCI, volume = "59", number = "4", pages = "154--157", year = "1983", CODEN = "PJAADT", ISSN = "0386-2194", MRclass = "33A30 (30D10)", MRnumber = "711323", MRreviewer = "R. A. Askey", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://projecteuclid.org/euclid.pja/1195515639", acknowledgement = ack-nhfb, fjournal = "Japan Academy. Proceedings. Series A. Mathematical Sciences", journal-URL = "http://projecteuclid.org/pja", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Borwein:1984:AGM, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", journal = j-SIAM-REVIEW, volume = "26", number = "3", pages = "351--366", month = jul, year = "1984", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1026073", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "65D20 (26A09)", MRnumber = "750454; 86d:65029", MRreviewer = "S. Conde", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "Compendex database; ftp://garbo.uwasa.fi/pc/doc-soft/fpbibl18.zip; garbo.uwasa.fi:/pc/doc-soft/fpbiblio.txt; http://epubs.siam.org/toc/siread/26/3; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://www.jstor.org/stable/2031275", abstract = "We produce a self contained account of the relationship between the Gaussian arithmetic--geometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for pi is one of the by-products.", acknowledgement = ack-nhfb # " and " # ack-nj, affiliationaddress = "Dalhousie Univ, Halifax, NS, Can", classification = "723; 921", fjournal = "SIAM Review. A Publication of the Society for Industrial and Applied Mathematics", journal-URL = "http://epubs.siam.org/sirev", journalabr = "SIAM Rev", keywords = "AGM (Arithmetic--Geometric Mean); arithmetic--geometric mean; calculation of pi; computational methods; elliptic functions; Iterative Methods; mathematical techniques; numerical mathematics", onlinedate = "July 1984", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Borwein:1984:GAG, author = "D. Borwein and P. B. Borwein", title = "A Generalized Arithmetic--Geometric Mean", journal = j-SIAM-REVIEW, volume = "26", number = "3", pages = "433--433", month = jul, year = "1984", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1026085", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:53:48 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/26/3; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", keywords = "AGM (Arithmetic--Geometric Mean)", onlinedate = "July 1984", } @TechReport{Borwein:1984:RCC, author = "J. M. Borwein and P. B. Borwein", title = "Reduced Complexity Calculation of Log", number = "DALTR 84-01", institution = "Department of Mathematics, Dalhousie University", address = "Halifax, NS, Canada", pages = "17", month = jan, year = "1984", bibdate = "Mon Nov 07 17:37:11 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", abstract = "Various reduced complexity methods for high precision computation of the logarithm are investigated.", acknowledgement = ack-nhfb, keywords = "analytic complexity; arithmetic--geometric mean; binary splitting; bit complexity; elliptic integrals; Fast Fourier Transform; logarithms; operational complexity; theta functions", remark = "Typescript, with 84-01 added by hand on cover page.", } @Article{Cox:1984:AGM, author = "David A. Cox", title = "The arithmetic--geometric mean of {Gauss}", journal = j-ENSEIGN-MATH-2, volume = "30", number = "3--4", pages = "275--330", year = "1984", CODEN = "ENMAAR", ISSN = "0013-8584 (print), 2309-4672 (electronic)", MRclass = "01A55 (11B83 11F03 14K25)", MRnumber = "767905", MRreviewer = "Bruce C. Berndt", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "L'Enseignement Math{\'e}matique. Revue Internationale. 2e S{\'e}rie", journal-URL = "http://www.e-periodica.ch/digbib/vollist?var=true&UID=ens-001", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Foster:1984:AHM, author = "D. M. E. Foster and G. M. Phillips", title = "The Arithmetic--Harmonic Mean", journal = j-MATH-COMPUT, volume = "42", number = "165", pages = "183--191", month = jan, year = "1984", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "40A99 (40A25)", MRnumber = "85j:40008", MRreviewer = "Amnon Jakimovski", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "C1120 (Mathematical analysis)", corpsource = "Maths. Inst., Univ. of St. Andrews, St. Andrews, Fife, UK", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "Archimedean process; arithmetic--harmonic; geometric mean; harmonic analysis; harmonic means; mean; sequences", treatment = "T Theoretical or Mathematical", } @Article{Schoen:1984:HGA, author = "Robert Schoen", title = "Harmonic, Geometric, and Arithmetic Means in Generalized {Fibonacci} Sequences", journal = j-FIB-QUART, volume = "22", number = "4", pages = "354--357", month = nov, year = "1984", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", MRclass = "11B39", MRnumber = "766313", bibdate = "Thu Oct 20 18:00:35 MDT 2011", bibsource = "http://www.fq.math.ca/22-4.html; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/22-4/schoen.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Cox:1985:GAG, author = "David A. Cox", title = "{Gauss} and the arithmetic--geometric mean", journal = j-NAMS, volume = "32", number = "2", pages = "147--151", year = "1985", CODEN = "AMNOAN", ISSN = "0002-9920 (print), 1088-9477 (electronic)", ISSN-L = "0002-9920", MRclass = "01A55 (11-03 14-03)", MRnumber = "779224", MRreviewer = "Willard Parker", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Notices of the American Mathematical Society", journal-URL = "http://www.ams.org/notices/", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Eddy:1985:BAG, author = "Roland H. Eddy", title = "Behold! {The} {Arithmetic-Geometric Mean Inequality}", journal = j-COLLEGE-MATH-J, volume = "16", number = "3", pages = "208--208", month = jun, year = "1985", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1985.11972881", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:09 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.1985.11972881", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Newman:1985:SVF, author = "D. J. Newman", title = "A simplified version of the fast algorithms of {Brent} and {Salamin}", journal = j-MATH-COMPUT, volume = "44", number = "169", pages = "207--210", month = jan, year = "1985", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "86e:65030", MRreviewer = "Walter Gautschi", bibdate = "Tue Oct 13 08:06:19 MDT 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); C4130 (Interpolation and function approximation)", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "exponential; fast algorithms; function approximation; function approximations; Gauss arithmetic--geometric process; pi", treatment = "T Theoretical or Mathematical", } @Article{Borwein:1986:MQC, author = "J. M. Borwein and P. B. Borwein", title = "More quadratically converging algorithms for $ \pi $", journal = j-MATH-COMPUT, volume = "46", number = "173", pages = "247--253", month = jan, year = "1986", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2008229", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "65D20", MRnumber = "815846; 87e:65014", MRreviewer = "M. M. Chawla", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", URL = "http://docserver.carma.newcastle.edu.au/1614/", acknowledgement = ack-nhfb, classcodes = "B0290F (Interpolation and function approximation); B0290Z (Other numerical methods); C4130 (Interpolation and function approximation); C4190 (Other numerical methods)", corpsource = "Dalhousie Univ., Halifax, NS, Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "arithmetic--geometric mean iteration; complete elliptic; convergence of numerical methods; Gauss--Legendre iteration; geometry; integrals; iterative; Legendre formula; methods; pi evaluation; quadratically converging algorithms", treatment = "T Theoretical or Mathematical", } @Article{Huda:1986:ESE, author = "S. Huda and Rahul Mukerjee", title = "{Edgeworth} series expansion for the distribution of the log of the ratio of arithmetic mean to geometric mean", journal = "Pakistan J. Statist.", volume = "2", number = "2", pages = "69--72", year = "1986", MRclass = "62E20", MRnumber = "909876", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Pakistan Journal of Statistics", } @Article{Razpet:1986:MAG, author = "Marko Razpet", title = "A method of arithmetic--geometric mean", journal = j-OBZORNIK-MAT-FIZ, volume = "33", number = "6", pages = "161--164", year = "1986", CODEN = "OBMFAY", ISSN = "0473-7446", MRclass = "65D30 (65D32)", MRnumber = "856885", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Dru{\v{s}}tvo Matematikov, Fizikov in Astronomov SRS. Obzornik za Matematiko in Fiziko", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Schattschneider:1986:PWA, author = "Doris Schattschneider", title = "Proof without Words: The Arithmetic Mean--Geometric Mean Inequality", journal = j-MATH-MAG, volume = "59", number = "1", pages = "11", year = "1986", CODEN = "MAMGA8", ISSN = "0025-570X", MRnumber = "1572599", bibdate = "Tue Aug 15 11:19:53 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2690011?origin=pubexport", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Zulauf:1986:SFE, author = "A. Zulauf", title = "Solution of a functional equation by use of weighted arithmetic--geometric means", journal = j-INDIAN-J-MATH, volume = "28", number = "1", pages = "49--56", year = "1986", CODEN = "IJOMAL", ISSN = "0019-5324", MRclass = "39B40", MRnumber = "868947", MRreviewer = "F. W. Carroll", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Indian Journal of Mathematics", journal-URL = "http://www.amsallahabad.org/ijm.html", zzbibdate = "Tue Aug 15 11:19:53 2017", } @Article{Alzer:1987:UZG, author = "Horst Alzer", title = "{{\"U}ber die Ungleichung zwischen dem geometrischen und dem arithmetischen Mittel}. ({German}) [{On} the inequality between the geometric and the arithmetic mean]", journal = "Quaestiones Math.", volume = "10", number = "4", pages = "351--356", year = "1987", ISSN = "0379-9468", MRclass = "26D15", MRnumber = "908677", MRreviewer = "L. Losonczi", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Quaestiones Mathematicae", language = "German", } @Unpublished{Borwein:1987:AGM, author = "Jonathan M. Borwein", title = "The arithmetic--geometric mean of {Gauss} and {Legendre}: An Excursion", day = "15", month = dec, year = "1987", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Canadian Mathematical Society, Coxeter--James Lecture, Vancouver, BC, Canada.", acknowledgement = ack-nhfb, } @Book{Borwein:1987:PAS, author = "Jonathan M. Borwein and Peter B. Borwein", title = "Pi and the {AGM}: a Study in Analytic Number Theory and Computational Complexity", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xv + 414", year = "1987", ISBN = "0-471-83138-7, 0-471-31515-X (paperback)", ISBN-13 = "978-0-471-83138-9, 978-0-471-31515-5 (paperback)", LCCN = "QA241 .B774 1987", MRclass = "11Y60 (68Q30)", MRnumber = "877728", MRreviewer = "H. London", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/bibnet/subjects/acc-stab-num-alg.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; z3950.loc.gov:7090/Voyager", series = "Canadian Mathematical Society series of monographs and advanced texts = Monographies et {\'e}tudes de la Soci{\'e}t{\'e} math{\'e}matique du Canada", acknowledgement = ack-nhfb, remark = "Chinese edition 1995.", subject = "Number theory; Computational complexity; Elliptic functions; Pi", xxURL = "http://www.loc.gov/catdir/description/wiley032/86015811.html; http://www.loc.gov/catdir/toc/onix02/86015811.html", } @Article{Burk:1987:NGL, author = "Frank Burk", title = "Notes: The Geometric, Logarithmic, and Arithmetic Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "94", number = "6", pages = "527--528", month = jun # "\slash " # jul, year = "1987", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2322844", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "DML", MRnumber = "1541119", bibdate = "Mon Jun 28 12:38:44 MDT 1999", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1980.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Cohen:1987:AGM, author = "Joel E. Cohen and Roger D. Nussbaum", title = "Arithmetic--geometric means of positive matrices", journal = j-MATH-PROC-CAMB-PHILOS-SOC, volume = "101", number = "2", pages = "209--219", year = "1987", CODEN = "MPCPCO", DOI = "https://doi.org/10.1017/S0305004100066561", ISSN = "0305-0041 (print), 1469-8064 (electronic)", ISSN-L = "0305-0041", MRclass = "15A51", MRnumber = "870592", MRreviewer = "Ray C. Shiflett", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Proceedings of the Cambridge Philosophical Society", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=PSP", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Minassian:1987:NAG, author = "Donald P. Minassian", title = "Notes: The Arithmetic--Geometric Mean Inequality Revisited: Elementary Calculus and Negative Numbers", journal = j-AMER-MATH-MONTHLY, volume = "94", number = "10", pages = "977--978", month = dec, year = "1987", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2322605", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26D15", MRnumber = "936057 (89e:26035)", MRreviewer = "G. A. Heuer", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1980.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Nelsen:1987:PWH, author = "Roger B. Nelsen", title = "Proof without Words: The Harmonic Mean--Geometric Mean--Arithmetic Mean--Root Mean Square Ineqality", journal = j-MATH-MAG, volume = "60", number = "3", pages = "158", year = "1987", CODEN = "MAMGA8", ISSN = "0025-570X", MRclass = "DML", MRnumber = "1572654", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2689561?origin=pubexport", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Reyssat:1987:AMA, author = "{\'E}ric Reyssat", title = "Approximation des moyennes arithm{\'e}tico--g{\'e}om{\'e}triques. ({French}) [{Approximation} of arithmetic--geometric means]", journal = j-ENSEIGN-MATH-2, volume = "33", number = "3--4", pages = "175--181", year = "1987", CODEN = "ENMAAR", ISSN = "0013-8584 (print), 2309-4672 (electronic)", MRclass = "11J82 (11B83 11F11)", MRnumber = "925983", MRreviewer = "John H. Loxton", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "L'Enseignement Math{\'e}matique. Revue Internationale. IIe S{\'e}rie", journal-URL = "http://www.e-periodica.ch/digbib/vollist?var=true&UID=ens-001", language = "French", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Seiffert:1987:WZG, author = "H.-J. Seiffert", title = "{Werte zwischen dem geometrischen und dem arithmetischen Mittel zweier Zahlen}. ({German}) [{Values} between the geometric and the arithmetic mean of two numbers]", journal = j-ELEM-MATH, volume = "42", number = "4", pages = "105--107", year = "1987", ISSN = "0013-6018 (print), 1420-8962 (electronic)", ISSN-L = "0013-6018", MRclass = "26D20", MRnumber = "896120", MRreviewer = "L. Losonczi", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Elemente der Mathematik. Revue de Math{\'e}matiques {\'E}l{\'e}mentaires. Rivista de Matematica Elementare", language = "German", } @Article{Sen:1987:WDA, author = "Pranab Kumar Sen", title = "What do the arithmetic, geometric and harmonic means tell us in length-biased sampling?", journal = j-STAT-PROB-LETT, volume = "5", number = "2", pages = "95--98", year = "1987", CODEN = "SPLTDC", DOI = "https://doi.org/10.1016/0167-7152(87)90062-9", ISSN = "0167-7152 (print), 1879-2103 (electronic)", ISSN-L = "0167-7152", MRclass = "62E10 (62E20)", MRnumber = "882342", MRreviewer = "Zhi-Dong Bai", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Statistics \& Probability Letters", journal-URL = "http://www.sciencedirect.com/science/journal/01677152", } @Article{Almkvist:1988:GLR, author = "Gert Almkvist and Bruce Berndt", title = "{Gauss}, {Landen}, {Ramanujan}, the Arithmetic--Geometric Mean, Ellipses, $ \pi $, and the {{\booktitle{Ladies Diary}}}", journal = j-AMER-MATH-MONTHLY, volume = "95", number = "7", pages = "585--608", month = aug # "\slash " # sep, year = "1988", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2323302", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "01A50 (01A55 01A60 33A25)", MRnumber = "966232; 89j:01028", MRreviewer = "R. A. Askey", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1980.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; JSTOR database", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Alzer:1988:UGA, author = "Horst Alzer", title = "{Ungleichungen f{\"u}r geometrische und arithmetische Mittelwerte}. ({German}) [{Inequalities} for geometric and arithmetic averages]", journal = "Nederl. Akad. Wetensch. Indag. Math.", volume = "50", number = "4", pages = "365--374", year = "1988", ISSN = "0019-3577, 0023-3358", MRclass = "26D20", MRnumber = "976521", MRreviewer = "Yisong Yang", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae", language = "German", } @Article{Askey:1988:BRP, author = "Richard Askey", title = "Reviews: {{\em Pi and the AGM}}, by {Jonathan M. Borwein and Peter B. Borwein}", journal = j-AMER-MATH-MONTHLY, volume = "95", number = "9", pages = "895--897", month = nov, year = "1988", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Sat Aug 13 18:09:13 MDT 2016", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1980.bib; JSTOR database", URL = "http://www.jstor.org/stable/2322925", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", keywords = "AGM (arithmetic--geometric mean)", } @Article{Berndt:1988:BRJ, author = "Bruce C. Berndt", title = "Book Review: {Jonathan M. Borwein and Peter B. Borwein, \booktitle{Pi and the AGM --- A Study of Analytic Number Theory and Computational Complexity}, Canadian Mathematical Society Series of Mono- graphs and Advanced Texts, Wiley, New York, 1987, xv + 414 pp., 24 cm. Price \$49.95}", journal = j-MATH-COMPUT, volume = "50", number = "181", pages = "352--354", month = jan, year = "1988", CODEN = "MCMPAF", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Sat Aug 13 18:09:13 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2007942", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Unpublished{Borwein:1988:AGMa, author = "Jonathan M. Borwein", title = "The arithmetic--geometric mean of {Gauss} and {Legendre}: An Excursion", day = "13", month = may, year = "1988", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Distinguished Lecturer Series, University of Delaware, Newark, DE, USA.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:1988:AGMb, author = "Jonathan M. Borwein", title = "The arithmetic--geometric mean of {Gauss} and {Legendre}: An Excursion", day = "14", month = jun, year = "1988", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, University of Newcastle, Newcastle, NSW, Australia.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:1988:AGMc, author = "Jonathan M. Borwein", title = "The arithmetic--geometric mean of {Gauss} and {Legendre}: An Excursion", day = "27", month = jun, year = "1988", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, University of New England, Armidale, NSW, Australia.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:1988:AGMd, author = "Jonathan M. Borwein", title = "The arithmetic--geometric mean of {Gauss} and {Legendre}: An Excursion", day = "27", month = jul, year = "1988", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, Auckland University, Auckland, New Zealand.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:1988:AGMe, author = "Jonathan M. Borwein", title = "The arithmetic--geometric mean of {Gauss} and {Legendre}: An Excursion", day = "12", month = sep, year = "1988", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, Macquarie University, Sydney, NSW, Australia.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:1988:BFD, author = "Jonathan M. Borwein", title = "{Borchardt}'s four-dimensional arithmetic--geometric mean", day = "14", month = sep, year = "1988", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Seminar, Macquarie University, Sydney, NSW, Australia.", acknowledgement = ack-nhfb, } @TechReport{Borwein:1988:CCJ, author = "J. M. Borwein and P. B. Borwein", title = "A Cubic Counterpart of {Jacobi}'s Identity and the {AGM}", type = "Report", institution = "Department of Mathematics, Statistics and Computing Science, Dalhousie University", address = "Halifax, NS B3H 3J5, Canada", pages = "20", day = "31", month = dec, year = "1988", bibdate = "Fri Nov 11 07:03:04 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", abstract = "We produce cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic--geometric mean iteration of Gauss and Legendre. The iteration in question is $ a_{n + 1} = (a_n + 2 b_n) / 3 $ and $ b_{n + 1} = \sqrt [3]{b_n((a_n^2 + a_n b_n + b_n^2) / 3)} $. The limit of this iteration is identified in terms of the hypergeometric function $_2 F_1 (1 / 3, 2 / 3; 1; \cdot)$ which supports a particularly simple cubic transformation.", acknowledgement = ack-nhfb, keywords = "$\pi$; arithmetic--geometric mean (AGM); cubic transformations; generalised elliptic functions; hypergeometric functions; mean iterations; theta functions", } @Article{Bost:1988:MAG, author = "Jean-Beno{\^{\i}}t Bost and Jean-Fran{\c{c}}ois Mestre", title = "Moyenne arithm{\'e}tico--g{\'e}om{\'e}trique et p{\'e}riodes des courbes de genre $1$ et $2$. ({French}) [{Arithmetic--geometric} mean and periods of the curves of genus $1$ and $2$]", journal = j-GAZ-MATH, volume = "38", number = "38", pages = "36--64", year = "1988", ISSN = "0224-8999", MRclass = "14K20 (11F03 11G05)", MRnumber = "970659", MRreviewer = "Reinhard B{\"o}lling", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Gazette des Math{\'e}maticiens", language = "French", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Hurley:1988:RCP, author = "Donal Hurley", title = "Recent computations of $ \pi $", journal = j-IRISH-MATH-SOC-BULL, volume = "21", number = "??", pages = "38--44", year = "1988", ISSN = "0791-5578", MRclass = "11Y60 (01A50 01A55 01A60 11-03)", MRnumber = "988289 (90e:11194)", MRreviewer = "Kenneth A. Jukes", bibdate = "Mon Apr 25 16:20:53 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, fjournal = "Irish Mathematical Society Bulletin", journal-URL = "https://www.maths.tcd.ie/pub/ims/bulletin/", keywords = "agm (arithmetic--geometric mean); Brent--Salamin algorithm (1976); Johann Dase (1824--1861); John Machin (1680--1752)", remark = "No issues before 1995 are available online at http://www.maths.tcd.ie/pub/ims/bulletin/index.php.", } @InProceedings{Kanada:1988:VMA, author = "Yasumasa Kanada", booktitle = "Proceedings of Supercomputing 88. Vol. II: Science and Applications", title = "Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of {$ \pi $} calculation", crossref = "Martin:1988:SPN", volume = "2", pages = "117--128", year = "1988", CODEN = "????", ISSN = "????", bibdate = "Sat Jul 16 16:53:44 MDT 2005", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", abstract = "More than 200 million decimal places of {$ \pi $} were calculated using an arithmetic geometric mean formula independently discovered by E. Salamin and R. P. Brent in 1976. Correctness of the calculation was verified through Borwein's quartic convergent formula developed in 1983. The computation took CPU times of 5 hours 57 minutes for the main calculation and 7 hours 30 minutes for the verification calculation on the HITAC S-820 model 80 supercomputer with 256 MB of main memory and 3 GB of high speed semiconductor storage, Extended Storage, to shorten I/O time.\par Computation was completed in 27th of January 1988. At that day two programs generated values up to $ 3 \times 2^{26} $, about 201 million. The two results agreed except for the last 21 digits. These results also agree with the 133,554,000 places of calculation of $ \pi $ which was done by the author in January 1987. Compare to the record in 1987, 50\% more decimal digits were calculated with about $ 1 / 6 $ of CPU time.\par Computation was performed with real arithmetic based vectorized Fast Fourier Transform (FFT) multiplier and newly vectorized multiple-precision add, subtract and (single word) constant multiplication programs. Vectorizations for the later cases were realized through first order linear recurrence vector instruction on the S-820. Details of the computation and statistical tests on the first 200 million digits of $ \pi - 3 $ are reported.", acknowledgement = ack-nhfb, classification = "C4190 (Other numerical methods); C7310 (Mathematics)", corpsource = "Comput. Centre, Tokyo Univ., Japan", keywords = "arithmetic geometric mean formula; Borwein's quartic convergent formula; fast Fourier transform; fast Fourier transforms; first order linear recurrence vector instruction; HITAC S-820 model 80 supercomputer; mathematics computing; multiple-precision arithmetic program; multiplier; parallel processing; pi calculation; S-820; vectorization", sponsororg = "IEEE; ACM SIGARCH", treatment = "P Practical", } @Article{Martins:1988:AGM, author = "J. S. Martins", title = "Arithmetic and geometric means, an application to {Lorentz} sequence spaces", journal = j-MATH-NACHR, volume = "139", pages = "281--288", year = "1988", CODEN = "MTMNAQ", DOI = "https://doi.org/10.1002/mana.19881390125", ISSN = "0025-584X", MRclass = "40A05 (26D20 40H05 46A45)", MRnumber = "978126", MRreviewer = "Christian Samuel", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematische Nachrichten", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1522-2616", } @Article{Montuchi:1988:BTE, author = "Paolo Montuchi and Warren Page", title = "Behold! {Two} Extremum Problems (and the {Arithmetic--Geometic Mean Inequality})", journal = j-COLLEGE-MATH-J, volume = "19", number = "4", pages = "347--347", month = sep, year = "1988", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1988.11973136", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:45 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.1988.11973136", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Nishiwada:1988:HSA, author = "Kimimasa Nishiwada", title = "A holomorphic structure of the arithmetic--geometric mean of {Gauss}", journal = j-PROC-JAPAN-ACAD-SER-A-MATH-SCI, volume = "64", number = "9", pages = "322--324", year = "1988", CODEN = "PJAADT", ISSN = "0386-2194", MRclass = "30B99", MRnumber = "979233", MRreviewer = "D. C. Russell", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://projecteuclid.org/euclid.pja/1195513088", acknowledgement = ack-nhfb, fjournal = "Japan Academy. Proceedings. Series A. Mathematical Sciences", journal-URL = "http://projecteuclid.org/pja", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Nussbaum:1988:AGM, author = "Roger D. Nussbaum and Joel E. Cohen", title = "The arithmetic--geometric mean and its generalizations for noncommuting linear operators", journal = j-ANN-SC-NORM-SUPER-PISA-CL-SCI, volume = "15", number = "2", pages = "239--308 (1989)", year = "1988", CODEN = "PSNAAI", ISSN = "0391-173x (print), 2036-2145 (electronic)", ISSN-L = "0391-173X", MRclass = "47B15 (26A18 47A60 47H07)", MRnumber = "1007399", MRreviewer = "T. Ando", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.numdam.org/item?id=ASNSP_1988_4_15_2_239_0", acknowledgement = ack-nhfb, fjournal = "Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Schaumberger:1988:GFA, author = "Norman Schaumberger", title = "A General Form of the Arithmetic-Geometric Mean Inequality via the Mean Value Theorem", journal = j-COLLEGE-MATH-J, volume = "19", number = "2", pages = "172--173", month = mar, year = "1988", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1988.11973110", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:41 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.1988.11973110", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Wang:1988:IRA, author = "Chung-Shin Wang and Gou Sheng Yang", title = "Inequalities related to the arithmetic and geometric means", journal = j-TAMKANG-J-MATH, volume = "19", number = "2", pages = "79--86", year = "1988", ISSN = "0049-2930 (print), 2073-9826 (electronic)", ISSN-L = "2073-9826", MRclass = "26D20", MRnumber = "996886", MRreviewer = "L. Losonczi", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Tamkang Journal of Mathematics", journal-URL = "http://journals.math.tku.edu.tw/index.php/TKJM", } @Article{Wimp:1988:BRP, author = "Jet Wimp", title = "Book Review: {{\booktitle{Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity}} (Jonathan M. Borwein and Peter B. Borwein)}", journal = j-SIAM-REVIEW, volume = "30", number = "3", pages = "530--533", month = sep, year = "1988", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1030128", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Aug 13 18:09:13 MDT 2016", bibsource = "http://epubs.siam.org/toc/siread/30/3; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", URL = "http://www.jstor.org/stable/2030735", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "September 1988", } @Article{Alzer:1989:RAM, author = "Horst Alzer", title = "A refinement of the arithmetic mean--geometric mean inequality", journal = "Rad. Mat.", volume = "5", number = "2", pages = "231--235", year = "1989", ISSN = "0352-6100", MRclass = "26D15", MRnumber = "1050892", MRreviewer = "E. Thandapani", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Radovi Matemati{\v{c}}ki", } @Article{Borwein:1989:MI, author = "J. M. Borwein and P. B. Borwein", title = "On the Mean Iteration $ (a, b) \leftarrow \big (\frac {a + 3b}{4}, \frac {\sqrt {ab} + b}{2} \big) $", journal = j-MATH-COMPUT, volume = "53", number = "187", pages = "311--326", month = jul, year = "1989", CODEN = "MCMPAF", DOI = "https://doi.org/10.2307/2008364", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", MRclass = "30D05 (33A25)", MRnumber = "968148, 90a:30075", MRreviewer = "Carl C. Cowen", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp1980.bib; JSTOR database", URL = "http://docserver.carma.newcastle.edu.au/1586/", acknowledgement = ack-nhfb, classcodes = "C4130 (Interpolation and function approximation)", corpsource = "Dept. of Math. Stat. and Comput. Sci., Dalhousie Univ., Halifax, NS, Canada", fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", keywords = "computation; convergence of numerical methods; converging process; iterative methods; iterative process; limit; mean iteration; nontrivial identifications; quadratically; symbolic; uniformizing parameters", treatment = "T Theoretical or Mathematical", } @Unpublished{Borwein:1989:PAG, author = "Jonathan M. Borwein", title = "Pi and the arithmetic--geometric mean", day = "14", month = apr, year = "1989", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, Rutger's University, New Brunswick, NJ, USA.", acknowledgement = ack-nhfb, } @Article{Henniart:1989:MAG, author = "Guy Henniart and Jean-Fran{\c{c}}ois Mestre", title = "Moyenne arithm{\'e}tico--g{\'e}om{\'e}trique $p$-adique. ({French}) [$p$-{Adic} arithmetic--geometric mean]", journal = j-C-R-ACAD-SCI-I, volume = "308", number = "13", pages = "391--395", year = "1989", CODEN = "CASMEI", ISSN = "0249-6291", ISSN-L = "0764-4442", MRclass = "11G07 (11F85 11Y99 14G20)", MRnumber = "992515", MRreviewer = "Glenn Stevens", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Comptes Rendus des S{\'e}ances de l'Acad{\'e}mie des Sciences. S{\'e}rie I. Math{\'e}matique", language = "French", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Nelsen:1989:RMS, author = "Roger B. Nelsen", title = "The Root Mean Square-Arithmetic Mean--Geometric Mean-Harmonic Mean Inequality", journal = j-COLLEGE-MATH-J, volume = "20", number = "3", pages = "231--231", month = may, year = "1989", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1989.11973236", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:54 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.1989.11973236", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Peetre:1989:GAG, author = "Jaak Peetre", title = "Generalizing the arithmetic geometric mean---a hapless computer experiment", journal = j-INT-J-MATH-MATH-SCI, volume = "12", number = "2", pages = "235--245", year = "1989", DOI = "https://doi.org/10.1155/S016117128900027X", ISSN = "0161-1712 (print), 1687-0425 (electronic)", ISSN-L = "0161-1712", MRclass = "26E99 (01A55 26-03)", MRnumber = "994905", MRreviewer = "Peter Borwein", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematics and Mathematical Sciences", journal-URL = "https://www.hindawi.com/journals/ijmms/", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Sala:1989:TJA, author = "Kenneth L. Sala", title = "Transformations of the {Jacobian} amplitude function and its calculation via the arithmetic--geometric mean", journal = j-SIAM-J-MATH-ANA, volume = "20", number = "6", pages = "1514--1528", month = nov, year = "1989", CODEN = "SJMAAH", DOI = "https://doi.org/10.1137/0520100", ISSN = "0036-1410 (print), 1095-7154 (electronic)", ISSN-L = "0036-1410", MRclass = "33A25 (42A16 70D99)", MRnumber = "1019316; 90j:33003", MRreviewer = "J. M. H. Peters", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMA/20/6; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmathana.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Mathematical Analysis", journal-URL = "http://epubs.siam.org/sima", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Schaumberger:1989:GI, author = "Norman Schaumberger", title = "The {AM-GM} Inequality via $ x^{1 / x} $", journal = j-COLLEGE-MATH-J, volume = "20", number = "4", pages = "320--320", month = sep, year = "1989", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.1989.11973249", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:50:55 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.1989.11973249", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Alzer:1990:CAM, author = "Horst Alzer", title = "A converse of the arithmetic mean--geometric mean inequality", journal = "Rev. Un. Mat. Argentina", volume = "36", pages = "146--151 (1992)", year = "1990", ISSN = "0041-6932", MRclass = "26D15", MRnumber = "1265703", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Revista de la Uni{\'o}n Matem{\'a}tica Argentina", } @Article{Alzer:1990:GGA, author = "Horst Alzer", title = "{{\"U}ber gewichtete geometrische und arithmetische Mittelwerte}. ({German}) [{On} overweighted geometric and arithmetic mean values]", journal = "Anz. {\"O}sterreich. Akad. Wiss. Math.-Natur. Kl.", volume = "127", pages = "33--36 (1991)", year = "1990", MRclass = "26D20", MRnumber = "1112640", MRreviewer = "P. S. Bullen", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "{\"O}sterreichische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Anzeiger", language = "German", } @Article{Alzer:1990:IAG, author = "Horst Alzer", title = "Inequalities for arithmetic, geometric and harmonic means", journal = j-BULL-LOND-MATH-SOC, volume = "22", number = "4", pages = "362--366", year = "1990", CODEN = "LMSBBT", DOI = "https://doi.org/10.1112/blms/22.4.362", ISSN = "0024-6093 (print), 1469-2120 (electronic)", ISSN-L = "0024-6093", MRclass = "26D15", MRnumber = "1058313", MRreviewer = "P. S. Bullen", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Bulletin of the London Mathematical Society", journal-URL = "http://blms.oxfordjournals.org/content/by/year", } @Article{Alzer:1990:LBD, author = "Horst Alzer", title = "A lower bound for the difference between the arithmetic and geometric means", journal = j-NIEUW-ARCHIEF-WISKUNDE-4, volume = "8", number = "2", pages = "195--197", year = "1990", CODEN = "NAWIA7", ISSN = "0028-9825", MRclass = "26D15", MRnumber = "1085159", MRreviewer = "P. S. Bullen", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Nieuw Archief voor Wiskunde. Vierde Serie", } @Article{Alzer:1990:SAM, author = "Horst Alzer", title = "Sharpenings of the arithmetic mean--geometric mean inequality", journal = j-CONG-NUM, volume = "75", pages = "63--66", year = "1990", ISSN = "0384-9864", MRclass = "26D20", MRnumber = "1069163", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Congressus Numerantium. A Conference Journal on Numerical Themes", remark = "Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989).", } @Article{Alzer:1990:WAG, author = "H. Alzer", title = "On weighted arithmetic, geometric and harmonic mean values", journal = "Glas. Mat. Ser. III", volume = "25(45)", number = "2", pages = "279--285", year = "1990", ISSN = "0017-095X", MRclass = "26D15", MRnumber = "1243732", MRreviewer = "Hiroshi Haruki", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Glasnik Matemati{\v{c}}ki. Serija III", } @Article{Chuan:1990:NIA, author = "Hao Zhi Chuan", title = "Note on the inequality of the arithmetic and geometric means", journal = j-PAC-J-MATH, volume = "143", number = "1", pages = "43--46", year = "1990", CODEN = "PJMAAI", ISSN = "0030-8730 (print), 1945-5844 (electronic)", ISSN-L = "0030-8730", MRclass = "26D15 (15A45)", MRnumber = "1047400", MRreviewer = "J. S{\'a}ndor", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://projecteuclid.org/euclid.pjm/1102646201", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Mathematics", journal-URL = "http://msp.org/pjm", } @Article{Nussbaum:1990:CPG, author = "Roger D. Nussbaum and Joel E. Cohen", title = "Convexity properties of generalizations of the arithmetic--geometric mean", journal = j-NUMER-FUNCT-ANAL-OPTIM, volume = "11", number = "1--2", pages = "33--44", year = "1990", CODEN = "NFAODL", DOI = "https://doi.org/10.1080/01630569008816359", ISSN = "0163-0563", MRclass = "26E05 (39B52 46G99 47H99)", MRnumber = "1058775", MRreviewer = "Sever S. Dragomir", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Numerical Functional Analysis and Optimization. An International Journal", journal-URL = "http://www.tandfonline.com/loi/lnfa20", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Sandor:1990:IAG, author = "J. S{\'a}ndor", title = "On the inequality of the arithmetic and geometric means", journal = "Bul. {\c{S}}tiin{\c{t}}. Inst. Politehn. Cluj-Napoca Ser. Mat. Mec. Apl. Construc. Ma{\c{s}}.", volume = "33", pages = "109--112", year = "1990", MRclass = "26D15", MRnumber = "1230744", MRreviewer = "B. Crstici", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Buletinul {\c{S}}tiin{\c{t}}ific al Institutului Politehnic Cluj-Napoca. Seria Matematic{\u{a}}, Mecanic{\v{a}} Aplicat\v a, Construc{\c{t}}ii de Ma{\c{s}}ini", } @Article{Alzer:1991:NAM, author = "Horst Alzer", title = "A note on the arithmetic mean-geometric mean inequality", journal = "Ann. Univ. Sci. Budapest. E{\"o}tv{\"o}s Sect. Math.", volume = "34", pages = "11--13 (1992)", year = "1991", ISSN = "0524-9007", MRclass = "26D15", MRnumber = "1161495", MRreviewer = "L. Losonczi", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Annales Universitatis Scientiarum Budapestinensis de Rolando E{\"o}tv{\"o}s Nominatae. Sectio Mathematica", } @Article{Borwein:1991:CCJ, author = "J. M. Borwein and P. B. Borwein", title = "A cubic counterpart of {Jacobi}'s identity and the {AGM}", journal = j-TRANS-AM-MATH-SOC, volume = "323", number = "2", pages = "691--701", month = feb, year = "1991", CODEN = "TAMTAM", DOI = "https://doi.org/10.2307/2001551", ISSN = "0002-9947 (print), 1088-6850 (electronic)", ISSN-L = "0002-9947", MRclass = "33C75 (11F11 11Y60 33C05)", MRnumber = "1010408", MRreviewer = "Bruce C. Berndt", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://docserver.carma.newcastle.edu.au/1578/; http://www.jstor.org/stable/2001551", acknowledgement = ack-nhfb, fjournal = "Transactions of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/tran/", } @Article{Bullett:1991:DAG, author = "Shaun Bullett", title = "Dynamics of the arithmetic--geometric mean", journal = j-TOPOLOGY, volume = "30", number = "2", pages = "171--190", year = "1991", CODEN = "TPLGAF", DOI = "https://doi.org/10.1016/0040-9383(91)90004-N", ISSN = "0040-9383 (print), 1879-3215 (electronic)", ISSN-L = "0040-9383", MRclass = "58F08 (58F23)", MRnumber = "1098912", MRreviewer = "Andrew Osbaldestin", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/004093839190004N", acknowledgement = ack-nhfb, fjournal = "Topology. An International Journal of Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/00409383", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Carlson:1991:IAL, author = "B. C. Carlson and M. Vuorinen", title = "Inequality of the {AGM} and the Logarithmic Mean", journal = j-SIAM-REVIEW, volume = "33", number = "4", pages = "655--655", month = "????", year = "1991", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1033141", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:54:57 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/33/4; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "December 1991", } @Article{Haruki:1991:NCA, author = "Hiroshi Haruki", title = "New characterizations of the arithmetic--geometric mean of {Gauss} and other well-known mean values", journal = j-PUBL-MATH-DEBRECEN, volume = "38", number = "3--4", pages = "323--332", year = "1991", CODEN = "PUMAAR", ISSN = "0033-3883 (print), 2064-2849 (electronic)", MRclass = "39B22 (26D99)", MRnumber = "1113240", MRreviewer = "Zsolt P{\'a}les", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Publicationes Mathematicae Debrecen", journal-URL = "http://publi.math.unideb.hu/", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Lingappaiah:1991:DRG, author = "G. S. Lingappaiah", title = "Distribution of the ratio of geometric mean to arithmetic mean in a sample from a two-piece double exponential distribution", journal = "Math. Balkanica (N.S.)", volume = "5", number = "1", pages = "76--80", year = "1991", ISSN = "0205-3217", MRclass = "62E15", MRnumber = "1136221", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematica Balkanica. New Series", } @Article{Szyszkowicz:1991:PGL, author = "Mieczyslaw Szyszkowicz", title = "Patterns generated by logical operators", journal = j-COMPUTERS-AND-GRAPHICS, volume = "15", number = "2", pages = "299--300", year = "1991", CODEN = "COGRD2", ISSN = "0097-8493 (print), 1873-7684 (electronic)", ISSN-L = "0097-8493", bibdate = "Fri Feb 07 10:57:32 1997", bibsource = "Compendex database; Graphics/imager/imager.91.bib; Graphics/siggraph/91.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/compgraph.bib", acknowledgement = ack-nhfb, classification = "723; 921", fjournal = "Computers \& Graphics", journal-URL = "http://www.sciencedirect.com/science/journal/00978493", journalabr = "Comput Graphics (Pergamon)", keywords = "Aesthetic Patterns; Arithmetic Geometric Mean Iterations; Computer Graphics; Computer Metatheory --- Formal Logic; Logical Operators; Mathematical Techniques --- Iterative Methods; Research", } @Article{Wazwaz:1991:MNM, author = "Abdul-Majid Wazwaz", title = "Modified numerical methods based on arithmetic and geometric means", journal = j-APPL-MATH-LETT, volume = "4", number = "3", pages = "49--52", year = "1991", CODEN = "AMLEEL", DOI = "https://doi.org/10.1016/0893-9659(91)90034-S", ISSN = "0893-9659 (print), 1873-5452 (electronic)", ISSN-L = "0893-9659", MRclass = "65L06 (65D32)", MRnumber = "1101874", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics Letters. An International Journal of Rapid Publication", journal-URL = "http://www.sciencedirect.com/science/journal/08939659", } @InCollection{Alzer:1992:IPA, author = "Horst Alzer", booktitle = "General inequalities, 6 ({Oberwolfach}, 1990)", title = "Inequalities for pseudo-arithmetic and geometric means", volume = "103", publisher = "Birkh{\"a}user, Basel", pages = "5--16", year = "1992", DOI = "https://doi.org/10.1007/978-3-0348-7565-3_2", MRclass = "26D15", MRnumber = "1212992", MRreviewer = "L. Losonczi", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Internat. Ser. Numer. Math.", acknowledgement = ack-nhfb, } @Article{Alzer:1992:SAM, author = "Horst Alzer", title = "A sharpening of the arithmetic mean-geometric mean inequality", journal = j-UTIL-MATH, volume = "41", pages = "249--252", year = "1992", CODEN = "UTMADA", ISSN = "0315-3681", MRclass = "26D15", MRnumber = "1162530", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Utilitas Mathematica. An International Journal of Discrete and Combinatorial Mathematics, and Statistical Design", } @Article{Braden:1992:IAL, author = "B. Braden and B. Danloy and F. Schmidt", title = "Inequality of the {AGM} and the Logarithmic Mean", journal = j-SIAM-REVIEW, volume = "34", number = "4", pages = "653--654", month = "????", year = "1992", CODEN = "SIREAD", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Fri Jun 21 11:25:02 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "December 1992", } @Article{Cohen:1992:RAG, author = "Joel E. Cohen and Thomas M. Liggett", title = "Random arithmetic--geometric means and random pi: observations and conjectures", journal = j-STOCH-PROC-APPL, volume = "41", number = "2", pages = "261--271", year = "1992", CODEN = "STOPB7", DOI = "https://doi.org/10.1016/0304-4149(92)90126-B", ISSN = "0304-4149 (print), 1879-209x (electronic)", ISSN-L = "0304-4149", MRclass = "60J05 (65D20 65U05)", MRnumber = "1164179", MRreviewer = "M. Iosifescu", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/030441499290126B", abstract = "Two random versions of the arithmetic--geometric mean of Gauss, Lagrange and Legendre are defined. Almost sure convergence and nondegeneracy are proved. These random arithmetic--geometric means in turn define two random versions of $ \pi $. Based on numerical simulations, inequalities and equalities are conjectured. A special case is proved. Further proofs are invited.", acknowledgement = ack-nhfb, fjournal = "Stochastic Processes and Their Applications", journal-URL = "http://www.sciencedirect.com/science/journal/03044149", keywords = "elliptic integrals; Markov processes with discrete parameter; nonlinear iteration; pi", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Unpublished{Dijkstra:1992:AMG, author = "Edsger W. Dijkstra", title = "The arithmetic mean and the geometric mean", month = oct, year = "1992", bibdate = "Mon Mar 16 08:14:00 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dijkstra-edsger-w.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Circulated privately.", URL = "http://www.cs.utexas.edu/users/EWD/ewd11xx/EWD1140.PDF", acknowledgement = ack-nhfb, filesize = "75 KB", oldlabel = "EWD:EWD1140", } @Article{Gauss:1992:AGM, author = "Karl F. Gauss", title = "La media aritm{\'e}tico geom{\'e}trica [The arithmetic--geometric mean] (de origene propietati busque generalis numerorum mediorum aritmeticorum--geometricorum)", journal = "Bol. Mat.", volume = "23", number = "1--2", pages = "69--79", year = "1992", CODEN = "BOMAD4", ISSN = "0120-0380 (print), 2357-6529 (electronic)", ISSN-L = "0120-0380", MRclass = "01A75 (01A55)", MRnumber = "1221411", MRreviewer = "Thomas Archibald", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Translated from the Latin original by Fabio Hernando Ortiz.", URL = "http://revistas.unal.edu.co/index.php/bolma/article/view/18204", acknowledgement = ack-nhfb, author-dates = "1777--1855", fjournal = "Bolet{\'\i}n de Matem{\'a}ticas", remark = "The Math Reviews commentary on this article reports some deficiencies and omissions in the translation, and notes that Gauss' Latin title was ``de origene propietatibusque generalibus numerorum mediorum aritmet. geometricorum''.", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Kittaneh:1992:NAG, author = "Fuad Kittaneh", title = "A note on the arithmetic--geometric-mean inequality for matrices", journal = j-LINEAR-ALGEBRA-APPL, volume = "171", pages = "1--8", year = "1992", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(92)90247-8", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A42", MRnumber = "1165442", MRreviewer = "Shao Kuan Li", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", zzbibdate = "Tue Aug 15 11:03:11 2017", } @InCollection{Roy:1992:CBA, author = "Dilip Roy and S. P. Mukherjee", booktitle = "Contributions to stochastics", title = "Characterizations based on arithmetic, geometric and harmonic means of failure rates", publisher = "Wiley, New York", pages = "178--185", year = "1992", MRclass = "62E10 (62N05)", MRnumber = "1223334", MRreviewer = "Eugenio Regazzini", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Todd:1992:BRB, author = "John Todd and Bill Braden and Bernard Danloy and Frank Schmidt", title = "Book Review: {{\booktitle{Inequality of the AGM and the Logarithmic Mean}} (B. C. Carlson and M. Vuorinen)}", journal = j-SIAM-REVIEW, volume = "34", number = "4", pages = "653--654", month = "????", year = "1992", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1034127", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:55:07 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/34/4; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "December 1992", } @Article{Bencze:1993:NPAa, author = "M. Bencze", title = "A new proof of the arithmetic--geometric mean inequality", journal = j-OCTOGON-MATH-MAG, volume = "1", number = "1", pages = "9--10", year = "1993", ISSN = "1222-5657 (print), 2248-1893 (electronic)", MRclass = "26-01 (26D20 40A99)", MRnumber = "1270849", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Octogon Mathematical Magazine", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Bencze:1993:NPAb, author = "Mihaly Bencze and Norman Schaumberger", title = "A New Proof of the Arithmetic--Geometric Mean Inequality", journal = j-MATH-MAG, volume = "66", number = "4", pages = "245", year = "1993", CODEN = "MAMGA8", ISSN = "0025-570X", MRclass = "DML", MRnumber = "1572972", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2690740?origin=pubexport", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Bhatia:1993:MMF, author = "Rajendra Bhatia and Chandler Davis", title = "More Matrix Forms of the Arithmetic--Geometric Mean Inequality", journal = j-SIAM-J-MAT-ANA-APPL, volume = "14", number = "1", pages = "132--136", month = jan, year = "1993", CODEN = "SJMAEL", DOI = "https://doi.org/10.1137/0614012", ISSN = "0895-4798 (print), 1095-7162 (electronic)", ISSN-L = "0895-4798", MRclass = "15A45 (15A60 47A63)", MRnumber = "1199551; 94b:15017", MRreviewer = "Ching-Tsuan Pan", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Matrix Analysis and Applications", journal-URL = "http://epubs.siam.org/simax", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Borwein:1993:HAA, author = "J. Borwein and P. Borwein and F. Garvan", title = "Hypergeometric analogues of the arithmetic--geometric mean iteration", journal = j-CONST-APPROX, volume = "9", number = "4", pages = "509--523", year = "1993", DOI = "https://doi.org/10.1007/BF01204654", ISSN = "0176-4276 (print), 1432-0940 (electronic)", ISSN-L = "0176-4276", MRclass = "33C05", MRnumber = "1237931", MRreviewer = "Bruce C. Berndt", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://docserver.carma.newcastle.edu.au/1556/; http://link.springer.com/article/10.1007/BF01204654", acknowledgement = ack-nhfb, fjournal = "Constructive Approximation. An International Journal for Approximations and Expansions", journal-URL = "http://link.springer.com/journal/365", zzbibdate = "Tue Aug 15 11:03:11 2017", } @Article{Borwein:1993:ICM, author = "Jonathan M. Borwein and Peter B. Borwein", title = "Inequalities for Compound Mean Iterations with Logarithmic Asymptotes", journal = j-J-MATH-ANAL-APPL, volume = "177", number = "2", pages = "572--582", year = "1993", CODEN = "JMANAK", DOI = "https://doi.org/10.1006/jmaa.1993.1278", ISSN = "0022-247X (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "33B99", MRnumber = "1231502", MRreviewer = "P. Anandani", bibdate = "Thu Aug 11 10:27:38 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://docserver.carma.newcastle.edu.au/1553/; http://www.sciencedirect.com/science/article/pii/S0022247X83712783", abstract = "We consider the compound means arising as limits from the arithmetic--geometric mean iteration and related iterations. Each of these iterations possesses a logarithmic asymptote. We show that these limit means satisfy very precise inequalities. These can be deduced in a quite uniform fashion from a `comparison lemma' for compound means.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", } @Article{Bullett:1993:RSP, author = "Shaun Bullett and Jaroslav Stark", title = "Renormalizing the Simple Pendulum", journal = j-SIAM-REVIEW, volume = "35", number = "4", pages = "631--640", month = dec, year = "1993", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1035140", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", MRclass = "70-01 (33E05 70K99)", MRnumber = "94i:70001", MRreviewer = "Coraci P. Malta", bibdate = "Sat Mar 29 09:55:16 MDT 2014", bibsource = "Compendex database; http://epubs.siam.org/toc/siread/35/4; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", abstract = "The authors present an elementary scheme for calculating the period of oscillation or rotation of a simple pendulum. It is based on the invariance of the complete elliptic integral of the first kind under the arithmetic--geometric mean iteration. They explain how this scheme can be interpreted as an example of renormalization, a technique with many recent applications in both physics and applied mathematics.", acknowledgement = ack-nhfb, affiliation = "Queen Mary and Westfield Coll", affiliationaddress = "London, Engl", classification = "921.2; 921.3; 921.6; 931.1; 943.1", fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", journalabr = "SIAM Rev", keywords = "Algorithms; Arithmetic--geometric mean; Elliptic integral; Equations of motion; Hamiltonian formalism; Integration; Iterative methods; Landen's transformation; Mathematical transformations; Oscillations; Pendulums; Renormalization; Rotation", onlinedate = "December 1993", } @Article{Dragomir:1993:ETR, author = "Sever Silvestru Dragomir", title = "Errata: ``{Two refinements of the arithmetic mean-geometric mean inequality'' [Nieuw Arch. Wisk.\ (4) {\bf 11} (1993), no. 1, 9--12; MR1220829 (94c:26025)]}", journal = j-NIEUW-ARCHIEF-WISKUNDE-4, volume = "11", number = "3", pages = "198", year = "1993", CODEN = "NAWIA7", ISSN = "0028-9825", MRclass = "26D15 (26B25)", MRnumber = "1251481", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Nieuw Archief voor Wiskunde. Vierde Serie", } @Article{Dragomir:1993:TRA, author = "Sever Silvestru Dragomir", title = "Two refinements of the arithmetic mean-geometric mean inequality", journal = j-NIEUW-ARCHIEF-WISKUNDE-4, volume = "11", number = "1", pages = "9--12", year = "1993", CODEN = "NAWIA7", ISSN = "0028-9825", MRclass = "26D15 (26B25)", MRnumber = "1220829", MRreviewer = "Hiroshi Haruki", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Nieuw Archief voor Wiskunde. Vierde Serie", } @Article{Hao:1993:RAG, author = "Zhi Chuan Hao", title = "A refinement of the arithmetic--geometric means inequality", journal = "J. Math. Res. Exposition", volume = "13", number = "1", pages = "84--88", year = "1993", CODEN = "SYPIET", ISSN = "1000-341X", MRclass = "26D15 (26D05)", MRnumber = "1211057", MRreviewer = "Hiroshi Haruki", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Research and Exposition", zzbibdate = "Tue Aug 15 11:03:11 2017", } @InCollection{Karmer:1993:MPC, author = "W. Karmer", title = "Multiple-precision computations with result verification", crossref = "Adams:1993:SCA", pages = "325--356", month = "????", year = "1993", bibdate = "Thu Dec 14 17:22:42 MST 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", abstract = "Multiple-precision real and interval modules for PASCAL-XSC have been developed. These modules are used to illustrate a variety of algorithms for the following purposes: multiple-precision evaluation of the function square root with maximum accuracy, the arithmetic--geometric mean iteration, different methods for the computation of a large number of digits of pi, the computation of elliptic integrals, the computation of guaranteed bounds for the natural logarithm, and the computation of e/sup pi / using a representation of this value by an infinite product. In general, enclosures for the desired values are computed. Due to the concept of overloading of functions and the operator concept of PASCAL-XSC the programs become clear and readable.", acknowledgement = ack-nhfb, affiliation = "Inst. fur Angewandte Math., Karlsruhe Univ., Germany", classification = "C5230 (Digital arithmetic methods); C6140D (High level languages); C7310 (Mathematics)", keywords = "Arithmetic--geometric mean iteration; Elliptic integrals; Enclosures; Function square root; Guaranteed bounds; Infinite product; Interval modules; Maximum accuracy; Multiple-precision evaluation; Natural logarithm; Operator concept; PASCAL-XSC; Result verification", pageswhole = "x + 612", pubcountry = "USA", thesaurus = "Digital arithmetic; High level languages; Mathematics computing; Pascal", } @InCollection{Kramer:1993:MPC, author = "Walter Kr{\"a}mer", booktitle = "Mathematics in Science and Engineering: Scientific Computing with Automatic Result Verification", title = "Multiple-Precision Computations with Result Verification", volume = "189", publisher = "Elsevier BV", address = "Amsterdam, The Netherlands", pages = "325--356", year = "1993", DOI = "https://doi.org/10.1016/s0076-5392(08)62851-9", bibdate = "Tue Mar 14 19:20:47 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, keywords = "arithmetic--geometric mean iteration; computation of $ e^\pi $; computation of a large number of digits of $\pi$; computation of elliptic integrals; computation of guaranteed bounds for the natural logarithm; interval arithmetic; PASCAL-XSC", } @Article{Mathias:1993:AGH, author = "Roy Mathias", title = "An arithmetic--geometric-harmonic mean inequality involving {Hadamard} products", journal = j-LINEAR-ALGEBRA-APPL, volume = "184", pages = "71--78", year = "1993", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(93)90370-4", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45 (26D15 47A63)", MRnumber = "1209383", MRreviewer = "Leo Livshits", bibdate = "Tue Aug 15 11:03:11 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @Article{Alzer:1994:NSA, author = "Horst Alzer", title = "Note on special arithmetic and geometric means", journal = "Comment. Math. Univ. Carolin.", volume = "35", number = "2", pages = "409--412", year = "1994", ISSN = "0010-2628 (print), 1213-7243 (electronic)", MRclass = "26D99 (26A99 40A05)", MRnumber = "1286588", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Commentationes Mathematicae Universitatis Carolinae", } @Article{Bencze:1994:NPW, author = "M. Bencze", title = "A new proof of the weighted arithmetic--geometric mean inequality", journal = j-OCTOGON-MATH-MAG, volume = "2", number = "1", pages = "17--18", year = "1994", ISSN = "1222-5657 (print), 2248-1893 (electronic)", MRclass = "26D20", MRnumber = "1303995", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Octogon Mathematical Magazine", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Unpublished{Dijkstra:1994:AAA, author = "Edsger W. Dijkstra", title = "The argument about the arithmetic mean and the geometric mean, heuristics included", month = jan, year = "1994", bibdate = "Mon Mar 16 08:14:00 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dijkstra-edsger-w.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Circulated privately.", URL = "http://www.cs.utexas.edu/users/EWD/ewd11xx/EWD1171.PDF", acknowledgement = ack-nhfb, filesize = "137 KB", oldlabel = "EWD:EWD1171", } @Article{Dzhaparidze:1994:SAI, author = "Kacha Dzhaparidze and Ren{\'e} H. P. Janssen", title = "A stochastic approach to an interpolation problem with applications to {Hellinger} integrals and arithmetic--geometric mean relationship", journal = j-CWI-QUARTERLY, volume = "7", number = "3", pages = "245--258", year = "1994", ISSN = "0922-5366", MRclass = "41A05 (41A55)", MRnumber = "1328044", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Centrum voor Wiskunde en Informatica. Centre for Mathematics and Computer Science. CWI Quarterly", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Furuta:1994:NAG, author = "Takayuki Furuta", title = "A note on the arithmetic--geometric mean inequality for every unitarily invariant matrix norm", journal = j-LINEAR-ALGEBRA-APPL, volume = "208--209", number = "??", pages = "223--228", year = "1994", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(94)90439-1", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A60", MRnumber = "1287348; 95f:15020", MRreviewer = "Frank Hansen", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala1990.bib", URL = "http://www.sciencedirect.com/science/article/pii/0024379594904391", abstract = "We integrate ten unitarily invariant matrix norm inequalities equivalent to the Heinz inequality.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795/", zzbibdate = "Tue Aug 15 10:35:36 2017", } @InCollection{Garvan:1994:CMI, author = "Frank Garvan", booktitle = "The {Rademacher} legacy to mathematics ({University} {Park}, {PA}, 1992)", title = "Cubic modular identities of {Ramanujan}, hypergeometric functions and analogues of the arithmetic--geometric mean iteration", volume = "166", publisher = "Amer. Math. Soc., Providence, RI", pages = "245--264", year = "1994", DOI = "https://doi.org/10.1090/conm/166/01633", MRclass = "39B12 (11F11 33C55)", MRnumber = "1284065", MRreviewer = "R. A. Askey", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Contemp. Math.", acknowledgement = ack-nhfb, zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Kedlaya:1994:PMA, author = "Kiran Kedlaya", title = "Proof of a mixed arithmetic-mean, geometric-mean inequality", journal = j-AMER-MATH-MONTHLY, volume = "101", number = "4", pages = "355--357", year = "1994", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2975630", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26D15", MRnumber = "1270962", MRreviewer = "F. Holland", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Kittaneh:1994:SOI, author = "Fuad Kittaneh", title = "On some operator inequalities", journal = j-LINEAR-ALGEBRA-APPL, volume = "208--209", number = "??", pages = "19--28", year = "1994", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(94)90427-8", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/0024379594904278", abstract = "For Hilbert-space operators $S$, $T$ with $S$ invertible and self-adjoint, Corach, Porta, and Recht recently proved that $ || S T S^{-1} + S^{-1} T S || \geq 2 || T ||$. A generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic--geometric-mean inequality. Some related inequalities are also discussed.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @Article{Liu:1994:GLA, author = "Zhi Guo Liu", title = "The geometric, logarithmic, and arithmetic mean inequalities in {$n$} variables", journal = "J. Chengdu Univ. Natur. Sci.", volume = "13", number = "1", pages = "37--41", year = "1994", ISSN = "1004-5422", MRclass = "26D15", MRnumber = "1399044", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Chengdu University. Natural Sciences. Chengdu Daxue Xuebao. Ziran Kexue Ban", } @Article{Nishiwada:1994:HSA, author = "Kimimasa Nishiwada", title = "Holomorphic structure of the arithmetic--geometric mean of {Gauss}. {II}", journal = j-PROC-JAPAN-ACAD-SER-A-MATH-SCI, volume = "70", number = "5", pages = "119--122", year = "1994", CODEN = "PJAADT", ISSN = "0386-2194", MRclass = "30B99 (40A05)", MRnumber = "1291164", MRreviewer = "D. C. Russell", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://projecteuclid.org/euclid.pja/1195511045", acknowledgement = ack-nhfb, fjournal = "Japan Academy. Proceedings. Series A. Mathematical Sciences", journal-URL = "http://projecteuclid.org/pja", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Sagae:1994:ULB, author = "Masahiko Sagae and Kunio Tanabe", title = "Upper and lower bounds for the arithmetic--geometric-harmonic means of positive definite matrices", journal = j-LIN-AND-MULT-ALGEBRA, volume = "37", number = "4", pages = "279--282", year = "1994", CODEN = "LNMLAZ", DOI = "https://doi.org/10.1080/03081089408818331", ISSN = "0308-1087 (print), 1563-5139 (electronic)", ISSN-L = "0308-1087", MRclass = "15A45 (15A48)", MRnumber = "1310971", MRreviewer = "Frank Hansen", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear and Multilinear Algebra", journal-URL = "http://www.tandfonline.com/loi/glma20", } @Article{Vamanamurthy:1994:IM, author = "M. K. Vamanamurthy and M. Vuorinen", title = "Inequalities for Means", journal = j-J-MATH-ANAL-APPL, volume = "183", number = "1", pages = "155--166", year = "1994", CODEN = "JMANAK", DOI = "https://doi.org/10.1006/jmaa.1994.1137", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X84711371", abstract = "A monotone form of L'Hospital's rule is obtained and applied to derive inequalities between the arithmetic--geometric mean of Gauss, the logarithmic mean, and Stolarsky's identric mean. Some related inequalities are given for complete elliptic integrals.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", } @Article{Alzer:1995:DII, author = "Horst Alzer", title = "On discrete inequalities involving arithmetic, geometric, and harmonic means", journal = "Rend. Istit. Mat. Univ. Trieste", volume = "27", number = "1-2", pages = "1--9 (1996)", year = "1995", ISSN = "0049-4704", MRclass = "26D15", MRnumber = "1421044", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Rendiconti dell'Istituto di Matematica dell'Universit{\`a} di Trieste. An International Journal of Mathematics", } @Article{Bencze:1995:NPA, author = "Mih{\'a}ly Bencze", title = "A new proof of the arithmetic--geometric pondered mean inequality", journal = j-OCTOGON-MATH-MAG, volume = "3", number = "1", pages = "16--17", year = "1995", ISSN = "1222-5657 (print), 2248-1893 (electronic)", MRclass = "26D15", MRnumber = "1361852", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Octogon Mathematical Magazine", } @Unpublished{Borwein:1995:CAD, author = "Jonathan M. Borwein", title = "The cubic {AGM} discovered", day = "26", month = oct, year = "1995", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Specialist Colloquium Lecture, University of Utrecht, Utrecht, The Netherlands.", acknowledgement = ack-nhfb, } @Article{Feng:1995:RAG, author = "Ci Huang Feng", title = "A refinement of the arithmetic--geometric mean inequality", journal = "J. Hangzhou Univ. Natur. Sci. Ed.", volume = "22", number = "3", pages = "222--225", year = "1995", CODEN = "HHHPD7", ISSN = "0253-3618", MRclass = "26D15", MRnumber = "1359614", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Hangzhou University. Natural Science Edition. Hangzhou Daxue Xuebao. Ziran Kexue Ban", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Horn:1995:NBH, author = "Roger A. Horn", title = "Norm Bounds for {Hadamard} Products and an Arithmetic--Geometric Mean Inequality for Unitarily Invariant Norms", journal = j-LINEAR-ALGEBRA-APPL, volume = "223--224", number = "1--3", pages = "355--361", day = "??", month = jul, year = "1995", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/0024-3795(94)00034-B", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45 (15A60 47A63)", MRnumber = "1340700; 96h:15020", MRreviewer = "Roy Mathias", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "http://www.elsevier.com/locate/laa; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala1990.bib", note = "Special issue honoring Miroslav Fiedler and Vlastimil Pt{\'a}k.", URL = "http://www.sciencedirect.com/science/article/pii/002437959400034B", abstract = "An arithmetic--geometric mean inequality for unitarily invariant norms and matrices, $ 2 || A \star X B || \leq || A A \star X + X B B \star || $, is an immediate consequence of a basic inequality for singular values of Hadamard products.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795/", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Lucht:1995:AGM, author = "Lutz G. Lucht", title = "On the arithmetic--geometric mean inequality", journal = j-AMER-MATH-MONTHLY, volume = "102", number = "8", pages = "739--740", month = oct, year = "1995", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2974645", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26D15", MRnumber = "1 357 492", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1990.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Mathias:1995:EAG, author = "Roy Mathias", title = "Erratum: ``{An} arithmetic--geometric-harmonic mean inequality involving {Hadamard} products'' [{Linear} {Algebra} {Appl}. {\bf 184} (1993), 71--78; {MR1209383} (94b:15019)]", journal = j-LINEAR-ALGEBRA-APPL, volume = "220", pages = "4", year = "1995", CODEN = "LAAPAW", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45 (26D15 47A63)", MRnumber = "1334561", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @Article{Matsuda:1995:IPM, author = "Takashi Matsuda", title = "An inductive proof of a mixed arithmetic--geometric mean inequality", journal = j-AMER-MATH-MONTHLY, volume = "102", number = "7", pages = "634--637", month = aug # "\slash " # sep, year = "1995", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2974561", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26D15", MRnumber = "1349877; 96h:26021", MRreviewer = "Zsolt P{\'a}les", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1990.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Nelsen:1995:PWA, author = "Roger B. Nelsen", title = "Proof without {Words}: {The} {Arithmetic}-{Logarithmic}-{Geometric} {Mean} {Inequality}", journal = j-MATH-MAG, volume = "68", number = "4", pages = "305", year = "1995", CODEN = "MAMGA8", ISSN = "0025-570X", MRclass = "DML", MRnumber = "1573115", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2690586?origin=pubexport", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Pecaric:1995:RSA, author = "Josip Pe{\v{c}}ari{\'c}", title = "On a recent sharpening of the arithmetic mean---geometric mean inequality", journal = j-UTIL-MATH, volume = "48", pages = "3--4", year = "1995", CODEN = "UTMADA", ISSN = "0315-3681", MRclass = "26D15", MRnumber = "1358587", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Utilitas Mathematica. An International Journal of Discrete and Combinatorial Mathematics, and Statistical Design", } @InProceedings{Sole:1995:A, author = "Patrick Sole", title = "{$ D_4 $}, {$ E_6 $}, {$ E_8 $} and the {AGM}", crossref = "Cohen:1995:AAA", volume = "948", pages = "448--455", year = "1995", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/3-540-60114-7_35", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Tue Mar 14 15:36:33 2017", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/lncs1995a.bib", abstract = "We derive Jacobi's quartic identity and the Borweins' cubic identity related to Ramanujan's quadratic modular equation on theta series by lattice enumerative methods. Both identities are instrumental in recent work of the Borweins on the Arithmetic Geometric Mean. Of great use are the constructions of the root lattices $ D_4 $ and $ E_6 $ by binary and ternary codes respectively. A third identity, equally due to the Borweins is also derived in relation to the root lattice $ E_8 $.", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Article{Alzer:1996:PAM, author = "Horst Alzer", title = "A proof of the arithmetic mean--geometric mean inequality", journal = j-AMER-MATH-MONTHLY, volume = "103", number = "7", pages = "585--585", month = aug # "\slash " # sep, year = "1996", CODEN = "AMMYAE", DOI = "https://doi.org/10.2307/2974672", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "26D15", MRnumber = "1 404 083", bibdate = "Wed Dec 3 17:17:33 MST 1997", bibsource = "http://www.jstor.org/journals/00029890.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly1990.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", zzbibdate = "Tue Aug 15 10:35:36 2017", } @TechReport{Borwein:1996:AGM, author = "Jonathan A. Borwein and Petr Lison{\v{e}}k and John A. Macdonald", title = "Arithmetic--Geometric Means Revisited", type = "Report", institution = inst-CECM, address = inst-CECM:adr, pages = "8", year = "1996", bibdate = "Thu Sep 01 10:39:15 2022", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://docserver.carma.newcastle.edu.au/4/", abstract = "We use Maple's {\tt gfun} library to study the limit formulae for a two-term recurrence (iteration) $ A G_N $, which in the case $ N = 2 $ specializes to the well-known Arithmetic-Geometric Mean iteration of Gauss. Our main aim 1s to independently rediscover and prove the limit formulae for two classical cases ($ N = 2, 3$) in a completely automated manner and to open the way for studying the remaining cases ($ N > 3$).", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Article{dalahu:1996:AIA, author = "Ba dalahu", title = "Applications of the inequality for arithmetic means and geometric means in nonlinear programming", journal = "Neimenggu Daxue Xuebao Ziran Kexue", volume = "27", number = "6", pages = "736--739", year = "1996", ISSN = "1000-1638", MRclass = "90C30 (65K05)", MRnumber = "1444556", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Neimenggu Daxue Xuebao. Ziran Kexue. Acta Scientiarum Naturalium Universitatis Neimenggu. Journal of Inner Mongolia University", } @InProceedings{Dijkstra:1996:AAA, author = "Edsger W. Dijkstra", title = "The argument about the arithmetic mean and the geometric mean, heuristics included", crossref = "Broy:1996:DPD", pages = "29--32", year = "1996", bibdate = "Wed Mar 18 12:28:57 2015", bibsource = "DBLP; http://dblp.uni-trier.de/db/conf/nato/dpd1996.html#Dijkstra96g; https://www.math.utah.edu/pub/bibnet/authors/d/dijkstra-edsger-w.bib; https://www.math.utah.edu/pub/mirrors/ftp.ira.uka.de/bibliography/Misc/DBLP/1996.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Unpublished{Dijkstra:1996:AGM, author = "Edsger W. Dijkstra", title = "The arithmetic and geometric means once more", month = feb, year = "1996", bibdate = "Mon Mar 16 08:14:00 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dijkstra-edsger-w.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Circulated privately.", URL = "http://www.cs.utexas.edu/users/EWD/ewd12xx/EWD1231.PDF", acknowledgement = ack-nhfb, filesize = "63 KB", oldlabel = "EWD:EWD1231", } @Article{Kanas:1996:UCC, author = "Stanis{\l}awa Kanas and Adam Lecko", title = "Univalence criteria connected with arithmetic and geometric means. {II}", journal = "Zeszyty Nauk. Politech. Rzeszowskiej Mat.", volume = "20", pages = "49--59", year = "1996", ISSN = "1232-7867", MRclass = "30C80", MRnumber = "1473957", MRreviewer = "William Ma", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka", } @InProceedings{Kaufmann:1996:IBMa, author = "Matt Kaufmann and Paolo Pecchiari", title = "Interaction with the {Boyer--Moore} Theorem Prover: a Tutorial Study Using the Arithmetic--Geometric Mean Theorem", crossref = "Zhang:1996:AMI", pages = "181--222", year = "1996", DOI = "https://doi.org/10.1007/978-94-009-1675-3_6", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-94-009-1675-3_6", acknowledgement = ack-nhfb, remark = "This chapter presents a formal proof with the Boyer--Moore Theorem Prover that the arithmetic mean of a sequence of natural numbers is greater than or equal to their geometric mean.", } @Article{Kaufmann:1996:IBMb, author = "Matt Kaufmann and Paolo Pecchiari", title = "Interaction with the {Boyer--Moore} theorem prover: a tutorial study using the arithmetic--geometric mean theorem", journal = j-J-AUTOM-REASON, volume = "16", number = "1--2", pages = "181--222", month = mar, year = "1996", CODEN = "JAREEW", DOI = "https://doi.org/10.1007/BF00244463", ISSN = "0168-7433 (print), 1573-0670 (electronic)", ISSN-L = "0168-7433", MRclass = "68T15 (03B35)", MRnumber = "1390909", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib", URL = "http://link.springer.com/article/10.1007/BF00244463", acknowledgement = ack-nhfb, ajournal = "J. Autom. Reason.", fjournal = "Journal of Automated Reasoning", journal-URL = "http://link.springer.com/journal/10817", zzbibdate = "Tue Aug 15 10:35:36 2017", } @InProceedings{Luther:1996:CAG, author = "Wolfram Luther and Werner Otten", title = "The Complex Arithmetic--Geometric Mean and Multiple-Precision Matrix Functions", crossref = "Alefeld:1996:SCV", pages = "52--58", year = "1996", MRclass = "65G10 (65H99)", MRnumber = "1394225", bibdate = "Mon May 20 06:32:10 MDT 2002", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; OCLC Proceedings database", acknowledgement = ack-nhfb, } @Article{Neuman:1996:TIA, author = "Edward Neuman", title = "Three Inequalities for the Arithmetic, Identric, and Geometric Means", journal = j-SIAM-REVIEW, volume = "38", number = "2", pages = "315--315", month = "????", year = "1996", CODEN = "SIREAD", DOI = "https://doi.org/10.1137/1038050", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Sat Mar 29 09:55:40 MDT 2014", bibsource = "http://epubs.siam.org/toc/siread/38/2; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "June 1996", } @Article{Sandor:1996:CIM, author = "J. S{\'a}ndor", title = "On Certain Inequalities for Means, {II}", journal = j-J-MATH-ANAL-APPL, volume = "199", number = "2", pages = "629--635", year = "1996", CODEN = "JMANAK", DOI = "https://doi.org/10.1006/jmaa.1996.0165", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X96901651", abstract = "A method based on sequences is applied to derive inequalities between the arithmetic--geometric mean of Gauss, the logarithmic mean, and certain other means.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", } @Article{Stefanski:1996:NAG, author = "L. A. Stefanski", title = "A note on the arithmetic--geometric-harmonic mean inequalities", journal = j-AMER-STAT, volume = "50", number = "3", pages = "246--247", year = "1996", CODEN = "ASTAAJ", DOI = "https://doi.org/10.2307/2684665", ISSN = "0003-1305 (print), 1537-2731 (electronic)", ISSN-L = "0003-1305", MRclass = "60E15", MRnumber = "1422074", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The American Statistician", journal-URL = "http://www.tandfonline.com/loi/utas20", } @Article{Alic:1997:AGH, author = "M. Ali{\'c} and B. Mond and J. Pe{\v{c}}ari{\'c} and V. Volenec", title = "The arithmetic--geometric--harmonic-mean and related matrix inequalities", journal = j-LINEAR-ALGEBRA-APPL, volume = "264", pages = "55--62", year = "1997", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/S0024-3795(96)00471-5", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45", MRnumber = "1465856", MRreviewer = "Yao Zhang", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @Article{Alic:1997:GAM, author = "M. Ali{\'c} and P. S. Bullen and J. E. Pe{\v{c}}ari{\'c} and V. Volenec", title = "On the geometric--arithmetic mean inequality for matrices", journal = "Math. Commun.", volume = "2", number = "2", pages = "125--128", year = "1997", ISSN = "1331-0623", MRclass = "15A45", MRnumber = "1612477", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Communications", } @Article{Alzer:1997:NRA, author = "Horst Alzer", title = "A new refinement of the arithmetic mean--geometric mean inequality", journal = j-ROCKY-MOUNTAIN-J-MATH, volume = "27", number = "3", pages = "663--667", year = "1997", CODEN = "RMJMAE", DOI = "https://doi.org/10.1216/rmjm/1181071887", ISSN = "0035-7596 (print), 1945-3795 (electronic)", ISSN-L = "0035-7596", MRclass = "26D20", MRnumber = "1490269", MRreviewer = "Hrvoje {\v{S}}iki{\'c}", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Rocky Mountain Journal of Mathematics", journal-URL = "http://projecteuclid.org/euclid.rmjm", } @Article{Ba:1997:SMA, author = "Dalahu Ba", title = "Some mixed arithmetic--geometric mean inequalities containing parameters and their applications", journal = "Neimenggu Daxue Xuebao Ziran Kexue", volume = "28", number = "6", pages = "731--734", year = "1997", CODEN = "NDZKEJ", ISSN = "1000-1638", MRclass = "26D15", MRnumber = "1620953", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Neimenggu Daxue Xuebao. Ziran Kexue. Acta Scientiarum Naturalium Universitatis Neimenggu. Journal of Inner Mongolia University", zzbibdate = "Tue Aug 15 10:35:36 2017", } @InCollection{Borwein:1997:AGMa, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", crossref = "Berggren:1997:PSB", pages = "537--552", year = "1997", DOI = "https://doi.org/10.1007/978-1-4757-2736-4_56", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Borwein:1984:AGM}.", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-2736-4_56", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Article{Borwein:1997:AGMb, author = "Jonathan A. Borwein and Petr Lison{\v{e}}k and John A. Macdonald", title = "Arithmetic--Geometric Means Revisited", journal = j-MAPLE-TECH-NEWS, volume = "4", number = "1", pages = "20--27", month = "Winter", year = "1997", ISSN = "1061-5733", ISSN-L = "1061-5733", bibdate = "Wed Jul 23 09:11:50 1997", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/maple-tech.bib", note = "Special issue on Maple in the mathematical sciences.", URL = "http://docserver.carma.newcastle.edu.au/4/", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", fjournal = "Maple technical newsletter", journal-URL = "http://web.mit.edu/maple/www/plibrary/mtn.html", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @InCollection{Cox:1997:AGM, author = "David A. Cox", title = "The Arithmetic--Geometric Mean of {Gauss}", crossref = "Berggren:1997:PSB", pages = "481--536", year = "1997", DOI = "https://doi.org/10.1007/978-1-4757-2736-4_55", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-2736-4_55", acknowledgement = ack-nhfb, } @Article{Dmitrieva:1997:FAB, author = "O. M. Dmitrieva and V. N. Maloz{\"e}mov", title = "On a fast algorithm based on arithmetic--geometric means", journal = j-ZH-VYCHISL-MAT-MAT-FIZ, volume = "37", number = "3", pages = "277--290", year = "1997", CODEN = "ZVMFAN", ISSN = "0044-4669", MRclass = "65B99 (11Y60 33E05 65D20)", MRnumber = "1452573", MRreviewer = "Gh. Adam", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Zhurnal Vychislitel\cprime no{\u\i} Matematiki i Matematichesko{\u\i} Fiziki. Rossi{\u\i}skaya Akademiya Nauk", journal-URL = "http://www.mathnet.ru/zvmmf", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Dragomir:1997:SMA, author = "S. S. Dragomir and D. Com{\v{a}}nescu and C. E. M. Pearce", title = "On some mappings associated with geometric and arithmetic means", journal = j-BULL-AUSTRAL-MATH-SOC, volume = "55", number = "2", pages = "299--309", year = "1997", CODEN = "ALNBAB", DOI = "https://doi.org/10.1017/S0004972700033967", ISSN = "0004-9727 (print), 1755-1633 (electronic)", ISSN-L = "0004-9727", MRclass = "26D10", MRnumber = "1438848", MRreviewer = "Zsolt P{\'a}les", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the Australian Mathematical Society", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=BAZ", } @Article{Gasharov:1997:SFT, author = "Vesselin Gasharov", title = "Symmetric functions and the theorem of the arithmetic and geometric means", journal = "J. Combin. Math. Combin. Comput.", volume = "25", pages = "91--95", year = "1997", ISSN = "0835-3026", MRclass = "05E05", MRnumber = "1480793", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Combinatorial Mathematics and Combinatorial Computing", } @Article{Murthy:1997:AMG, author = "Amarnath Murthy", title = "On the arithmetic mean geometric mean inequality--another two short proofs", journal = "Math. Ed. (Siwan)", volume = "31", number = "2", pages = "118--120", year = "1997", ISSN = "0047-6269", MRclass = "26D15", MRnumber = "1463401", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematics Education", } @Article{Nishiwada:1997:AAG, author = "Kimimasa Nishiwada", title = "Algorithm of the arithmetic--geometric mean and its complex limits", journal = j-HOKKAIDO-MATH-J, volume = "26", number = "3", pages = "541--564", year = "1997", CODEN = "HMAJDN", DOI = "https://doi.org/10.14492/hokmj/1351258265", ISSN = "0385-4035", MRclass = "11F99 (11F06 33E05 40A05)", MRnumber = "1483461", MRreviewer = "D. C. Russell", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Hokkaido Mathematical Journal", journal-URL = "http://projecteuclid.org/hokmj", zzbibdate = "Tue Aug 15 10:35:36 2017", } @InCollection{Pecaric:1997:AMG, author = "J. Pe{\v{c}}ari{\'c} and B. Mond", booktitle = "General inequalities, 7 ({Oberwolfach}, 1995)", title = "The arithmetic mean---the geometric mean and related matrix inequalities", volume = "123", publisher = "Birkh{\"a}user, Basel", pages = "77--91", year = "1997", MRclass = "15A45 (26D15 47A63)", MRnumber = "1457271", MRreviewer = "I. Gavrea", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Internat. Ser. Numer. Math.", acknowledgement = ack-nhfb, } @Article{Pecaric:1997:NPA, author = "Josip Pe{\v{c}}ari{\'c} and Sanja Varo{\v{s}}anec", title = "A new proof of the arithmetic mean---the geometric mean inequality", journal = j-J-MATH-ANAL-APPL, volume = "215", number = "2", pages = "577--578", year = "1997", CODEN = "JMANAK", DOI = "https://doi.org/10.1006/jmaa.1997.5616", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "26D15", MRnumber = "1490770", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", } @InCollection{Salamin:1997:CUA, author = "Eugene Salamin", title = "Computation of $ \pi $ Using Arithmetic--Geometric Mean", crossref = "Berggren:1997:PSB", pages = "418--423", year = "1997", DOI = "https://doi.org/10.1007/978-1-4757-2736-4_46", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-2736-4_46", acknowledgement = ack-nhfb, } @Article{Seiffert:1997:TIA, author = "H. J. Seiffert", title = "Three inequalities for the arithmetic, identric, and geometric means", journal = j-SIAM-REVIEW, volume = "39", number = "2", pages = "330--332", month = "????", year = "1997", CODEN = "SIREAD", ISSN = "0036-1445 (print), 1095-7200 (electronic)", ISSN-L = "0036-1445", bibdate = "Fri Jun 21 11:25:02 MDT 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamreview.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Review", journal-URL = "http://epubs.siam.org/sirev", onlinedate = "June 1997", xxnote = "Check title word: identric??", } @Article{Bencze:1998:NPA, author = "Mih{\'a}ly Bencze", title = "A new proof of the arithmetic--geometric mean inequality", journal = j-OCTOGON-MATH-MAG, volume = "6", number = "1", pages = "49--50", year = "1998", ISSN = "1222-5657 (print), 2248-1893 (electronic)", MRclass = "26D15", MRnumber = "1630953", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Octogon Mathematical Magazine", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Book{Borwein:1998:PAS, author = "Jonathan M. Borwein and Peter B. Borwein", title = "Pi and the {AGM}: A study in analytic number theory and computational complexity", publisher = pub-WILEY, address = pub-WILEY:adr, pages = "xvi + 414", year = "1998", ISBN = "0-471-31515-X", ISBN-13 = "978-0-471-31515-5", MRclass = "11Y60 (11B65 68Q25)", MRnumber = "1641658", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Reprint of the 1987 original.", series = "Canadian Mathematical Society Series of Monographs and Advanced Texts, 4", acknowledgement = ack-nhfb, } @Article{Dragomir:1998:IAG, author = "Sever S. Dragomir", title = "The improvement of arithmetic--geometric inequality for weighted means", journal = "Ranchi Univ. Math. J.", volume = "29", pages = "11--19 (1999)", year = "1998", ISSN = "0079-9602", MRclass = "26D15", MRnumber = "1755103", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Ranchi University Mathematical Journal", } @InCollection{Kanas:1998:UCC, author = "S. Kanas and A. Lecko", booktitle = "Transform methods \& special functions, {Varna} '96", title = "Univalence criteria connected with arithmetic and geometric means. {I}", publisher = "Bulgarian Acad. Sci., Sofia", pages = "201--209", year = "1998", MRclass = "30C80", MRnumber = "1667743", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Kosaki:1998:AGM, author = "Hideki Kosaki", title = "Arithmetic--Geometric Mean and Related Inequalities for Operators", journal = j-J-FUNCT-ANAL, volume = "156", number = "2", pages = "429--451", year = "1998", CODEN = "JFUAAW", DOI = "https://doi.org/10.1006/jfan.1998.3258", ISSN = "0022-1236 (print), 1096-0783 (electronic)", ISSN-L = "0022-1236", MRclass = "47A63 (47A30)", MRnumber = "1636964", MRreviewer = "Takayuki Furuta", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S002212369893258X", abstract = "In recent years certain arithmetic--geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on. We first point out that they are direct consequences of integral expressions of relevant operators. Furthermore we obtain related new inequalities (Theorems 4, 5, and 6) based on our current approach.", acknowledgement = ack-nhfb, fjournal = "Journal of Functional Analysis", journal-URL = "http://www.sciencedirect.com/science/journal/00221236", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Maligranda:1998:WHI, author = "Lech Maligranda", title = "Why {H{\"o}lder's Inequality} should be called {Rogers' Inequality}", journal = j-MATH-INEQUAL-APPL, volume = "1", number = "1", pages = "69--83", month = "????", year = "1998", ISSN = "1331-4343 (print), 1848-9966 (electronic)", ISSN-L = "1331-4343", bibdate = "Fri Feb 15 16:18:07 2013", bibsource = "http://mia.ele-math.com/; https://www.math.utah.edu/pub/tex/bib/agm.bib", ZMnumber = "0889.26001", acknowledgement = ack-nhfb, ajournal = "Math. Inequal. Appl.", fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", keywords = "H{\"o}lder inequality; Cauchy inequality; Jensen inequality; history; biographies", ZMclass = "26-03 (Historical (real functions)); 26D15 (Inequalities for sums, series and integrals of real functions); 01A50 (Mathematics in the 18th century); 01A60 (Mathematics in the 20th century); 01A70 (Biographies, obituaries, personalia, bibliographies)", ZMreviewer = "Johann Acz{\'e}l (Waterloo/Ontario)", } @Article{Sole:1998:LCA, author = "P. Sol{\'e} and P. Loyer", title = "{$ U_n $} Lattices, Construction {$B$}, and {AGM} Iterations", journal = j-EUR-J-COMB, volume = "19", number = "2", pages = "227--236", month = feb, year = "1998", CODEN = "EJOCDI", DOI = "https://doi.org/10.1006/eujc.1997.0185", ISSN = "0195-6698 (print), 1095-9971 (electronic)", ISSN-L = "0195-6698", bibdate = "Tue Mar 14 17:10:52 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "European Journal of Combinatorics", } @Article{Toader:1998:SMV, author = "Gh. Toader", title = "Some Mean Values Related to the Arithmetic--Geometric Mean", journal = j-J-MATH-ANAL-APPL, volume = "218", number = "2", pages = "358--368", year = "1998", CODEN = "JMANAK", DOI = "https://doi.org/10.1006/jmaa.1997.5766", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "26D15 (39B22)", MRnumber = "1606799", MRreviewer = "Zsolt P{\'a}les", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X97957668", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Alzer:1999:SIA, author = "Horst Alzer", title = "Some inequalities for arithmetic and geometric means", journal = j-PROC-R-SOC-EDINB-SECT-A-MATH, volume = "129", number = "2", pages = "221--228", year = "1999", CODEN = "PEAMDU", DOI = "https://doi.org/10.1017/S0308210500021326", ISSN = "0308-2105 (print), 1473-7124 (electronic)", ISSN-L = "0308-2105", MRclass = "26D15", MRnumber = "1686698", MRreviewer = "Wolfram Koepf", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the Royal Society of Edinburgh. Section A. Mathematics", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=PRM", } @TechReport{Boldo:1999:CRE, author = "Sylvie Boldo", title = "Calcul rapide et exact de fonctions {\'e}l{\'e}mentaires en pr{\'e}cision arbitraire par la moyenne arithm{\'e}tico-g{\'e}om{\'e}trique. ({French}) [Rapid and exact computation of elementary functions in arbitrary precision by the arithmetic--geometric mean]", type = "Report", institution = "INRIA, Projet Spaces, LORIA, Campus Scientifique", address = "B.P. 239, 54506 Vandoeuvre-l{\`e}s-Nancy Cedex, France", pages = "29", year = "1999", bibdate = "Tue Nov 23 11:00:03 2004", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Under the direction of Paul Zimmermann.", URL = "http://perso.ens-lyon.fr/sylvie.boldo/doc/mpfr.ps", acknowledgement = ack-nhfb, language = "French", } @Article{Cass:1999:BRP, author = "Peter Cass", title = "Book Review: {{\booktitle{Pi and the AGM}}}", journal = j-MATH-GAZ, volume = "83", number = "497", pages = "334--335", month = jul, year = "1999", CODEN = "MAGAAS", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Sat Aug 13 18:09:13 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/3619084", acknowledgement = ack-nhfb, fjournal = "Mathematical Gazette", journal-URL = "http://www.m-a.org.uk/jsp/index.jsp?lnk=620", } @Article{Donagi:1999:AGM, author = "Ron Donagi and Ron Livn{\'e}", title = "The arithmetic--geometric mean and isogenies for curves of higher genus", journal = j-ANN-SC-NORM-SUPER-PISA-CL-SCI, volume = "28", number = "2", pages = "323--339", year = "1999", CODEN = "PSNAAI", ISSN = "0391-173x (print), 2036-2145 (electronic)", ISSN-L = "0391-173X", MRclass = "14H40 (14K02)", MRnumber = "1736231", MRreviewer = "Arnaud Beauville", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.numdam.org/item?id=ASNSP_1999_4_28_2_323_0", acknowledgement = ack-nhfb, fjournal = "Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Dragomir:1999:CAM, author = "S. S. Dragomir", title = "Counterparts of arithmetic mean--geometric mean--harmonic mean inequality", journal = "Studia Univ. Babe{\c{s}}-Bolyai Math.", volume = "44", number = "4", pages = "37--42", year = "1999", ISSN = "0252-1938", MRclass = "26D15", MRnumber = "1989064", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Universitatis Babe{\c{s}}-Bolyai. Studia. Mathematica", } @Article{Fournier:1999:EGA, author = "Richard Fournier", title = "Extensions of the geometric--arithmetic means inequality to a disc of the complex plane", journal = j-MATH-INEQUAL-APPL, volume = "2", number = "1", pages = "19--24", year = "1999", DOI = "https://doi.org/10.7153/mia-02-03", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "30C45 (26D15)", MRnumber = "1667789", MRreviewer = "Wolfram Koepf", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Hiai:1999:CVM, author = "Fumio Hiai and Hideki Kosaki", title = "Comparison of Various Means for Operators", journal = j-J-FUNCT-ANAL, volume = "163", number = "2", pages = "300--323", year = "1999", CODEN = "JFUAAW", DOI = "https://doi.org/10.1006/jfan.1998.3375", ISSN = "0022-1236 (print), 1096-0783 (electronic)", ISSN-L = "0022-1236", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022123698933754", abstract = "For Hilbert space operators $H$, $K$, $X$ with $ H, K \geq 0$ the norm inequality $ |||H^{1 / 2} X K^{1 / 2}||| \leq (1 / 2) |||H X + X K|||$ is known, where ||| \cdot ||| is an arbitrary unitarily invariant norm. A refinement of this arithmetic--geometric mean inequality is studied. Similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean.", acknowledgement = ack-nhfb, fjournal = "Journal of functional analysis", journal-URL = "http://www.sciencedirect.com/science/journal/00221236", } @Article{Joseph:1999:AMG, author = "James E. Joseph and Myung H. Kwack", title = "The arithmetic-mean--geometric-mean inequality derived from closed polynomial functions", journal = "Missouri J. Math. Sci.", volume = "11", number = "2", pages = "103--106", year = "1999", ISSN = "0899-6180", MRclass = "26D15", MRnumber = "1694273", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Missouri Journal of Mathematical Sciences", } @Article{Kittaneh:1999:SNI, author = "Fuad Kittaneh", title = "Some norm inequalities for operators", journal = j-CAN-MATH-BULL, volume = "42", number = "1", pages = "87--96", month = mar, year = "1999", CODEN = "CMBUA3", DOI = "https://doi.org/10.4153/CMB-1999-010-6", ISSN = "0008-4395 (print), 1496-4287 (electronic)", ISSN-L = "0008-4395", MRclass = "47A30, 47B10, 47B15, 47B20", bibdate = "Thu Sep 8 10:22:25 MDT 2011", bibsource = "http://cms.math.ca/cmb/v42/; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/canmathbull.bib", abstract = "Let $ A_i $, $ B_i $ and $ X_i $ $ (i = 1, 2, \dots, n) $ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $ [0, \infty)$ which satisfy the relation $ f(t)g(t) = t$ for all $t$ in $ [0, \infty)$, then $$ \Biglvert \, \Bigl | \sum^n_{i = 1} A^*_i X_i B_i \Bigr |^r \, \Bigrvert^2 \leq \Biglvert \Bigl (\sum^n_{i = 1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl (\sum^n_{i = 1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert $$ for every $ r > 0$ and for every unitarily invariant norm. This result improves some known Cauchy--Schwarz type inequalities. Norm inequalities related to the arithmetic--geometric mean inequality and the classical Heinz inequalities are also obtained.", acknowledgement = ack-nhfb, ams-subject-primary = "47A30, 47B10, 47B15, 47B20", fjournal = "Canadian Mathematical Bulletin. Bulletin Canadien de Math{\'e}matiques", journal-URL = "http://cms.math.ca/cmb/", journalabbrev = "CMB", keywords = "arithmetic--geometric mean inequality; Cauchy--Schwarz inequality; Heinz inequality; positive operator; Unitarily invariant norm", refnum = "7284", } @InCollection{Latala:1999:EBG, author = "Rafa{\l} Lata{\l}a", booktitle = "Convex geometric analysis ({Berkeley}, {CA}, 1996)", title = "On the equivalence between geometric and arithmetic means for log-concave measures", volume = "34", publisher = "Cambridge Univ. Press, Cambridge", pages = "123--127", year = "1999", DOI = "https://doi.org/10.2977/prims/1195144757", MRclass = "60E15 (60B11)", MRnumber = "1665584", MRreviewer = "Pawel Hitczenko", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Math. Sci. Res. Inst. Publ.", acknowledgement = ack-nhfb, } @Article{Sampedro:1999:DIP, author = "J. Cruz Sampedro and M. Tetlalmatzi Montiel", title = "A direct inductive proof of the geometric mean--arithmetic mean inequality", journal = "Miscel{\'a}nea Mat.", volume = "28", pages = "11--15", year = "1999", ISSN = "1665-5478", MRclass = "26D15", MRnumber = "1823254", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Miscel{\'a}nea Matem{\'a}tica", } @Article{Sandor:1999:AGM, author = "J{\'o}zsef S{\'a}ndor", title = "On the arithmetic--geometric mean of {Gauss}", journal = j-OCTOGON-MATH-MAG, volume = "7", number = "1", pages = "108--115", year = "1999", ISSN = "1222-5657 (print), 2248-1893 (electronic)", MRclass = "26D15", MRnumber = "1730039", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Octogon Mathematical Magazine", zzbibdate = "Tue Aug 15 10:35:36 2017", } @Article{Wang:1999:SLT, author = "Liqiu Wang", title = "Second law of thermodynamics and arithmetic-mean--geometric-mean inequality", journal = "Internat. J. Modern Phys. B", volume = "13", number = "21-22", pages = "2791--2793", year = "1999", DOI = "https://doi.org/10.1142/S0217979299002678", ISSN = "0217-9792 (print), 1793-6578 (electronic)", MRclass = "80A10 (26D15)", MRnumber = "1713415", bibdate = "Tue Aug 15 10:35:36 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Modern Physics B", } @Article{Alsina:2000:PWA, author = "Claudi Alsina", title = "Proof without Words: The Arithmetic--Geometric Mean Inequality for Three Positive Numbers", journal = j-MATH-MAG, volume = "73", number = "2", pages = "97", year = "2000", CODEN = "MAMGA8", ISSN = "0025-570X", MRnumber = "1573443", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2691079?origin=pubexport", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Bhatia:2000:NMA, author = "Rajendra Bhatia and Fuad Kittaneh", title = "Notes on matrix arithmetic--geometric mean inequalities", journal = j-LINEAR-ALGEBRA-APPL, volume = "308", number = "1--3", pages = "203--211", day = "15", month = mar, year = "2000", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/S0024-3795(00)00048-3", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45 (15A42 15A60 47A30)", MRnumber = "1751140", MRreviewer = "Yao Zhang", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "http://www.elsevier.com/locate/laa; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2000.bib", URL = "http://www.elsevier.nl/gej-ng/10/30/19/125/25/36/abstract.html; http://www.elsevier.nl/gej-ng/10/30/19/125/25/36/article.pdf; http://www.sciencedirect.com/science/article/pii/S0024379500000483", abstract = "For positive semi-definite $ n \times n $ matrices, the inequality $ 4 ||| A B ||| \leq ||| (A + B)^2 ||| $ is shown to hold for every unitarily invariant norm. The connection of this with some other matrix arithmetic--geometric mean inequalities and trace inequalities is discussed.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Arithmetic--geometric mean; Majorisation; Singular values; Unitarily invariant norms", } @InCollection{Borwein:2000:AGM, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", crossref = "Berggren:2000:PSB", pages = "537--552", year = "2000", DOI = "https://doi.org/10.1007/978-1-4757-3240-5_56", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Borwein:1984:AGM}.", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-3240-5_56", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @InCollection{Cox:2000:AGM, author = "David A. Cox", title = "The Arithmetic--Geometric Mean of {Gauss}", crossref = "Berggren:2000:PSB", pages = "481--536", year = "2000", DOI = "https://doi.org/10.1007/978-1-4757-3240-5_55", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-3240-5_55", acknowledgement = ack-nhfb, } @Article{Hao:2000:CIA, author = "Zhi Chuan Hao", title = "A combinatorial inequality for the arithmetic and geometric means", journal = "Guizhou Shifan Daxue Xuebao Ziran Kexue Ban", volume = "18", number = "1", pages = "28--31", year = "2000", ISSN = "1004-5570", MRclass = "26D15", MRnumber = "1748759", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Guizhou Shifan Daxue Xuebao. Ziran Kexue Ban. Journal of Guizhou Normal University. Natural Sciences", } @InCollection{Kwon:2000:AGM, author = "Ern Gun Kwon and Kwang Ho Shon", booktitle = "Finite or infinite dimensional complex analysis ({Fukuoka}, 1999)", title = "On the arithmetic--geometric mean inequality", volume = "214", publisher = "Dekker, New York", pages = "233--235", year = "2000", MRclass = "26D15", MRnumber = "1771322", MRreviewer = "L. Losonczi", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Lecture Notes in Pure and Appl. Math.", acknowledgement = ack-nhfb, } @Article{Kwon:2000:ATB, author = "E. G. Kwon and S. T. Lee and K. H. Shon", title = "An additional term between arithmetic mean and geometric mean", journal = "Bull. Korean Math. Soc.", volume = "37", number = "2", pages = "285--289", year = "2000", ISSN = "1015-8634", MRclass = "26D15 (28A35)", MRnumber = "1757495", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the Korean Mathematical Society", } @InCollection{Salamin:2000:CUA, author = "Eugene Salamin", title = "Computation of $ \pi $ Using Arithmetic--Geometric Mean", crossref = "Berggren:2000:PSB", publisher = pub-SV, address = pub-SV:adr, pages = "418--423", year = "2000", DOI = "https://doi.org/10.1007/978-1-4757-3240-5_46", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-3240-5_46", acknowledgement = ack-nhfb, } @Article{Sury:2000:AGM, author = "B. Sury", title = "The arithmetico--geometric mean of {Gauss}: How to find the perimeter of an ellipse", journal = j-RESONANCE, volume = "5", number = "8", pages = "72--83", month = aug, year = "2000", CODEN = "RESOFE", DOI = "https://doi.org/10.1007/bf02837938", ISSN = "0971-8044 (print), 0973-712X (electronic)", bibdate = "Tue Mar 14 15:45:02 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Resonance", journal-URL = "http://link.springer.com/journal/12045", } @Article{Wachspress:2000:EEF, author = "E. L. Wachspress", title = "Evaluating elliptic functions and their inverses", journal = j-COMPUT-MATH-APPL, volume = "39", number = "3--4", pages = "131--136", month = feb, year = "2000", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/S0898-1221(99)00339-9", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Wed Mar 1 21:49:06 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122199003399", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "arithmetic--geometric mean (AGM)", } @Article{Yang:2000:AGM, author = "Ling Ou Yang", title = "Arithmetic--geometric mean inequalities for trace of operators", journal = "Math. Theory Appl. (Changsha)", volume = "20", number = "3", pages = "117--120", year = "2000", ISSN = "1006-8074", MRclass = "47A64 (47A63)", MRnumber = "1806708", MRreviewer = "Frank Hansen", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Theory and Applications", } @Article{Bracken:2001:AGM, author = "Paul Bracken", title = "An arithmetic--geometric mean inequality", journal = j-EXPO-MATH, volume = "19", number = "3", pages = "273--279", year = "2001", DOI = "https://doi.org/10.1016/S0723-0869(01)80006-2", ISSN = "0723-0869 (print), 1878-0792 (electronic)", ISSN-L = "0723-0869", MRclass = "26D15", MRnumber = "1852077", MRreviewer = "Antal Bege", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0723086901800062", abstract = "Several integrals which are related to the arithmetic--geometric mean are developed and proved in a very elementary way. These results can be used to prove a known inequality which relates this mean to the logarithmic mean.", acknowledgement = ack-nhfb, fjournal = "Expositiones Mathematicae", journal-URL = "http://www.sciencedirect.com/science/journal/07230869", } @Article{Dobbs:2001:PAG, author = "David E. Dobbs", title = "A proof of the arithmetic--geometric mean inequality using non-{Euclidean} geometry", journal = j-INT-J-MATH-EDU-SCI-TECH, volume = "32", number = "5", pages = "778--782", year = "2001", CODEN = "IJMEBM", DOI = "https://doi.org/10.1080/002073901753124655", ISSN = "0020-739x (print), 1464-5211 (electronic)", ISSN-L = "0020-739X", MRclass = "26D15 (51M10)", MRnumber = "1862675", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Education in Science and Technology", journal-URL = "http://www.tandfonline.com/loi/tmes20", } @Article{Ekart:2001:NGA, author = "Anik{\'o} Ek{\'a}rt and S. Z. N{\'e}meth", title = "A noncontinuous generalization of the arithmetic--geometric mean", journal = j-APPL-MATH-COMP, volume = "124", number = "2", pages = "261--279", day = "25", month = oct, year = "2001", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/S0096-3003(00)00098-9", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", MRclass = "40A05", MRnumber = "1857690", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "http://www.elsevier.com/locate/issn/00963003; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib", URL = "http://www.elsevier.com/gej-ng/10/9/12/113/31/36/abstract.html; http://www.sciencedirect.com/science/article/pii/S0096300300000989", abstract = "The notions of prickly set, scalar and vectorial mean are defined. A noncontinuous generalization of the arithmetic--geometric mean is given, by considering the recursion xn+1=F(xn), where F:C \to C is a vectorial mean and C is a closed prickly subset of Rm. The convergence of this recursion is proved and it is shown that the limit is contained in the diagonal of C. If F is continuous, it is deduced that the limit of the recursion is a continuous function of the initial value x=x0. Denoting the limit by F\infty(x) it is proved that if F is monotone, then F\infty it is also monotone (where the monotonicity is considered with respect to the closed cone R+m).", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "Arithmetic--geometric mean; F-mean; Prickly set; Scalar mean; Vectorial mean", } @Article{Nakamura:2001:AAA, author = "Yoshimasa Nakamura", title = "Algorithms associated with arithmetic, geometric and harmonic means and integrable systems", journal = j-J-COMPUT-APPL-MATH, volume = "131", number = "1--2", pages = "161--174", day = "1", month = jun, year = "2001", CODEN = "JCAMDI", DOI = "https://doi.org/10.1016/S0377-0427(00)00316-2", ISSN = "0377-0427 (print), 1879-1778 (electronic)", ISSN-L = "0377-0427", MRclass = "65P40 (37N30 39A10)", MRnumber = "1835710", bibdate = "Sat Feb 25 12:45:18 MST 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0377042700003162", acknowledgement = ack-nhfb, fjournal = "Journal of Computational and Applied Mathematics", journal-URL = "http://www.sciencedirect.com/science/journal/03770427", } @Article{Sury:2001:LA, author = "B. Sury", title = "Logarithm and {AGM}", journal = j-RESONANCE, volume = "6", number = "11", pages = "85--86", month = nov, year = "2001", CODEN = "RESOFE", DOI = "https://doi.org/10.1007/bf02868248", ISSN = "0971-8044 (print), 0973-712X (electronic)", bibdate = "Tue Mar 14 15:35:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Resonance", journal-URL = "http://link.springer.com/journal/12045", } @Article{Yang:2001:AGM, author = "Hansheng Yang and Heng Yang", title = "The Arithmetic--Geometric Mean Inequality and the Constant $e$", journal = j-MATH-MAG, volume = "74", number = "4", pages = "321--323", year = "2001", CODEN = "MAMGA8", ISSN = "0025-570X", MRnumber = "1573553", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.jstor.org/stable/2691107?origin=pubexport", acknowledgement = ack-nhfb, fjournal = "Mathematics Magazine", journal-URL = "http://www.maa.org/pubs/mathmag.html", } @Article{Alzer:2002:AMG, author = "Horst Alzer and Stephan Ruscheweyh", title = "The arithmetic mean--geometric mean inequality for complex numbers", journal = "Analysis (Munich)", volume = "22", number = "3", pages = "277--283", year = "2002", DOI = "https://doi.org/10.1524/anly.2002.22.3.277", ISSN = "0174-4747", MRclass = "30A10 (26E60)", MRnumber = "1938378", MRreviewer = "Arcadii Z. Grinshpan", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Analysis. International Mathematical Journal of Analysis and its Applications", } @Article{Charzynski:2002:IBA, author = "Zygmunt Karol Charzy{\'n}ski", title = "On an inequality between an arithmetic mean and a geometric mean", journal = "Zeszyty Nauk. Politech. Rzeszowskiej Mat.", volume = "26", pages = "165--171", year = "2002", ISSN = "1232-7867", MRclass = "26D15 (26E60)", MRnumber = "1949601", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka", } @Article{Disch:2002:CPV, author = "Burkhard Disch", title = "Computing present values by the {AGM}", journal = "{Bl{\"a}tter der DGVFM}", volume = "25", number = "4", pages = "831--849", month = oct, year = "2002", DOI = "https://doi.org/10.1007/bf02809119", bibdate = "Tue Mar 14 15:31:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Gaudry:2002:CCS, author = "Pierrick Gaudry", booktitle = "{International Conference on the Theory and Application of Cryptology and Information Security: ASIACRYPT 2002: Advances in Cryptology --- ASIACRYPT 2002}", title = "A Comparison and a Combination of {SST} and {AGM} Algorithms for Counting Points of Elliptic Curves in Characteristic $2$", volume = "2501", publisher = pub-SV, address = pub-SV:adr, pages = "311--327", year = "2002", CODEN = "LNCSD9", DOI = "https://doi.org/10.1007/3-540-36178-2_20", ISSN = "0302-9743 (print), 1611-3349 (electronic)", ISSN-L = "0302-9743", bibdate = "Tue Mar 14 15:33:17 2017", bibsource = "http://link.springer-ny.com/link/service/series/0558/tocs/t2501.htm; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; https://www.math.utah.edu/pub/tex/bib/lncs2002e.bib", series = "Lecture Notes in Computer Science", URL = "http://link.springer.de/link/service/series/0558/bibs/2501/25010311.htm; http://link.springer.de/link/service/series/0558/papers/2501/25010311.pdf", acknowledgement = ack-nhfb, fjournal = "Lecture Notes in Computer Science", journal-URL = "http://link.springer.com/bookseries/558", } @Article{Georgakis:2002:IAG, author = "Constantine Georgakis", title = "On the inequality for the arithmetic and geometric means", journal = j-MATH-INEQUAL-APPL, volume = "5", number = "2", pages = "215--218", year = "2002", DOI = "https://doi.org/10.7153/mia-05-23", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "26D15", MRnumber = "1899088", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Xie:2002:SGA, author = "Ziqing Xie", title = "On the Summation of Generalized Arithmetic--Geometric Trigonometric Series", journal = j-FIB-QUART, volume = "40", number = "2", pages = "128--135", month = may, year = "2002", CODEN = "FIBQAU", ISSN = "0015-0517", ISSN-L = "0015-0517", bibdate = "Thu Oct 20 18:03:33 MDT 2011", bibsource = "http://www.fq.math.ca/40-2.html; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fibquart.bib", URL = "http://www.fq.math.ca/Scanned/40-2/xie.pdf", acknowledgement = ack-nhfb, ajournal = "Fib. Quart", fjournal = "The Fibonacci Quarterly. Official Organ of the Fibonacci Association", journal-URL = "http://www.fq.math.ca/", } @Unpublished{Borwein:2003:ACFa, author = "Jonathan M. Borwein", title = "The {AGM} Continued Fraction of {Ramanujan}", day = "31", month = jul, year = "2003", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "CECM Day 2003, Simon Fraser University, Burnaby, BC, Canada.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:2003:ACFb, author = "Jonathan M. Borwein", title = "The {AGM} Continued Fraction of {Ramanujan}", day = "16", month = sep, year = "2003", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "First Plenary Lecture, First Congress of the Mathematical Society of South East Europe (MASSE{\'E}), Borovets, Bulgaria.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:2003:ACFc, author = "Jonathan M. Borwein", title = "The {AGM} Continued Fraction of {Ramanujan}", day = "14", month = oct, year = "2003", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, Reed College, OR, USA.", acknowledgement = ack-nhfb, } @Article{Gluskin:2003:NGA, author = "E. Gluskin and V. Milman", title = "Note on the Geometric--Arithmetic Mean Inequality", journal = j-LECT-NOTES-MATH, volume = "1807", pages = "131--135", year = "2003", CODEN = "LNMAA2", DOI = "https://doi.org/10.1007/978-3-540-36428-3_11", ISBN = "3-540-00485-8 (print), 3-540-36428-5 (e-book)", ISBN-13 = "978-3-540-00485-1 (print), 978-3-540-36428-3 (e-book)", ISSN = "0075-8434 (print), 1617-9692 (electronic)", ISSN-L = "0075-8434", MRclass = "46B20 (46B09)", MRnumber = "2083393", MRreviewer = "Niels J\o rgen Nielsen", bibdate = "Fri May 9 19:07:18 MDT 2014", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/lnm2000.bib", URL = "http://link.springer.com/chapter/10.1007/978-3-540-36428-3_11/", acknowledgement = ack-nhfb, book-DOI = "https://doi.org/10.1007/b10415", book-URL = "http://www.springerlink.com/content/978-3-540-36428-3", fjournal = "Lecture Notes in Mathematics", journal-URL = "http://link.springer.com/bookseries/304", xxpages = "130--135", } @Article{Guo:2003:IMR, author = "Bai-Ni Guo and Feng Qi", title = "Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with arbitrary difference", journal = j-TAMKANG-J-MATH, volume = "34", number = "3", pages = "261--270", year = "2003", ISSN = "0049-2930 (print), 2073-9826 (electronic)", ISSN-L = "2073-9826", MRclass = "26D07 (26A48 26D15 26E60)", MRnumber = "2001922", MRreviewer = "Attila Gil{\'a}nyi", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Tamkang Journal of Mathematics", journal-URL = "http://journals.math.tku.edu.tw/index.php/TKJM", } @Article{Katsuura:2003:GAG, author = "Hidefumi Katsuura", title = "Generalizations of the Arithmetic--Geometric Mean Inequality and a Three Dimensional Puzzle", journal = j-COLLEGE-MATH-J, volume = "34", number = "4", pages = "280--282", month = sep, year = "2003", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.2003.11922018", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 09:53:32 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.1080/07468342.2003.11922018", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "30 Jan 2018", } @Article{Knockaert:2003:BUB, author = "Luc Knockaert", title = "Best upper bounds based on the arithmetic--geometric mean inequality", journal = j-ARCH-INEQUAL-APPL, volume = "1", number = "1", pages = "85--90", year = "2003", ISSN = "1542-6149", MRclass = "15A42 (15A12 60E15)", MRnumber = "1992268", MRreviewer = "E. Seneta", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Archives of Inequalities and Applications. An International Journal for Theory and Applications", } @Article{Mercer:2003:RAG, author = "Peter R. Mercer", title = "Refined arithmetic, geometric and harmonic mean inequalities", journal = j-ROCKY-MOUNTAIN-J-MATH, volume = "33", number = "4", pages = "1459--1464", year = "2003", CODEN = "RMJMAE", DOI = "https://doi.org/10.1216/rmjm/1181075474", ISSN = "0035-7596 (print), 1945-3795 (electronic)", ISSN-L = "0035-7596", MRclass = "26D15 (26E60)", MRnumber = "2052499", MRreviewer = "J. Horv{\'a}th", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Rocky Mountain Journal of Mathematics", journal-URL = "http://projecteuclid.org/euclid.rmjm", } @Article{Monhor:2003:AGM, author = "D. Monhor", title = "The arithmetic--geometric mean and the elliptic mean error", journal = j-ACTA-GEOD-GEOPHYS-HU, volume = "38", number = "1", pages = "??--??", month = feb, year = "2003", CODEN = "AGGHFW", DOI = "https://doi.org/10.1556/AGeod.38.2003.1.8", ISSN = "1217-8977 (print), 1587-1037 (electronic)", ISSN-L = "1217-8977", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/article/10.1556/AGeod.38.2003.1.8", acknowledgement = ack-nhfb, fjournal = "Acta Geodaetica et Geophysica Hungarica", journal-URL = "http://www.akademiai.com/loi/074", remark = "Journal archive contains only years 2001 to date.", } @Article{Qi:2003:IMRa, author = "Feng Qi", title = "Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with unit difference", journal = j-AUSTRALIAN-MATH-SOC-GAZ, volume = "30", number = "3", pages = "142--147", year = "2003", ISSN = "0311-0729 (print), 1326-2297 (electronic)", ISSN-L = "0311-0729", MRclass = "26D07 (26E60)", MRnumber = "1988519", MRreviewer = "Attila Gil{\'a}nyi", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Australian Mathematical Society. Gazette", journal-URL = "http://www.austms.org.au/gazette", } @Article{Qi:2003:IMRb, author = "Feng Qi", title = "Inequalities and monotonicity of the ratio of the geometric means of a positive arithmetic sequence with unit difference", journal = j-INT-J-MATH-EDU-SCI-TECH, volume = "34", number = "4", pages = "601--607", year = "2003", CODEN = "IJMEBM", DOI = "https://doi.org/10.1080/0020739031000149010", ISSN = "0020-739x (print), 1464-5211 (electronic)", ISSN-L = "0020-739X", MRclass = "11B75 (11B25 26D15)", MRnumber = "1998815", MRreviewer = "Shiro Ando", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Education in Science and Technology", journal-URL = "http://www.tandfonline.com/loi/tmes20", } @Article{Rooin:2003:AIB, author = "Jamal Rooin", title = "{AGM} inequality with binomial expansion", journal = j-ELEM-MATH, volume = "58", number = "3", pages = "115--117", month = aug, year = "2003", DOI = "https://doi.org/10.1007/s00017-003-0188-x", ISSN = "0013-6018 (print), 1420-8962 (electronic)", ISSN-L = "0013-6018", bibdate = "Tue Mar 14 15:30:54 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Elemente der Mathematik", } @Article{Roy:2003:CBW, author = "Dilip Roy", title = "Characterization of a bivariate {Weibull} distribution based on arithmetic, geometric and harmonic means of failure rates", journal = "J. Appl. Statist. Sci.", volume = "12", number = "3", pages = "191--199", year = "2003", ISSN = "1067-5817", MRclass = "62N05 (62E10 62H05)", MRnumber = "2038808", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Statistical Science", } @Misc{Tkachev:2003:EFI, author = "Vladimir G. Tkachev", title = "Elliptic functions: Introduction course", howpublished = "Web lecture notes.", day = "25", month = nov, year = "2003", bibdate = "Wed Mar 15 08:43:21 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf", acknowledgement = ack-nhfb, tableofcontents = "Chapter 1. Elliptic integrals and Jacobi's theta functions / 5 \\ 1.1. Elliptic integrals and the AGM: real case / 5 \\ 1.1.3. The arithmetic--geometric mean iteration / 7 \\ 1.2. Lemniscates and elastic curves / 11 \\ 1.3. Euler's addition theorem / 18 \\ 1.4. Theta functions: preliminaries 5 / 24 \\ Chapter 2. General theory of doubly periodic functions / 31 \\ 2.1. Preliminaries / 31 \\ 2.2. Periods of analytic functions / 33 \\ 2.3. Existence of doubly periodic functions / 36 \\ 2.4. Liouville's theorems / 38 \\ 2.5. The Weierstrass function $\wp(z)$ / 43 \\ 2.6. Modular forms / 51 \\ Bibliography / 61", } @InCollection{Borwein:2004:AGMa, author = "J. M. Borwein and P. B. Borwein", title = "The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions", crossref = "Berggren:2004:PSB", pages = "537--552", year = "2004", DOI = "https://doi.org/10.1007/978-1-4757-4217-6_56", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", note = "Reprint of \cite{Borwein:1984:AGM}.", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", } @Unpublished{Borwein:2004:RACa, author = "Jonathan M. Borwein", title = "{Ramanujan}'s {AGM} Continued Fractions and Dynamics: the real case", day = "4", month = mar, year = "2004", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Colloquium, Mathematics Department, Dalhousie University, Halifax, NS, Canada.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:2004:RACb, author = "Jonathan M. Borwein", title = "{Ramanujan}'s {AGM} Continued Fractions and Dynamics: the complex case", day = "10", month = mar, year = "2004", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Analysis Seminar, Mathematics Department, Dalhousie University, Halifax, NS, Canada.", acknowledgement = ack-nhfb, } @Unpublished{Borwein:2004:RACc, author = "Jonathan M. Borwein", title = "{Ramanujan}'s {AGM} Continued Fractions and Dynamics", day = "27", month = aug, year = "2004", bibdate = "Tue Aug 16 10:19:46 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Workshop on Analytic and Computational Number Theory, August 23--27, Dalhousie University, Halifax, NS, Canada.", acknowledgement = ack-nhfb, } @Article{Borwein:2004:RAFa, author = "J. Borwein and R. Crandall and G. Fee", title = "On the {Ramanujan} {AGM} fraction. {I}. {The} real-parameter case", journal = j-EXP-MATH, volume = "13", number = "3", pages = "275--285", month = "????", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2004.10504540", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", MRclass = "11J70 (11A55 33C05 40A15)", MRnumber = "2103326 (2005g:11126)", MRreviewer = "James G. Mc Laughlin", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "http://projecteuclid.org/euclid.em; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib", URL = "http://docserver.carma.newcastle.edu.au/27/; http://projecteuclid.org/euclid.em/1103749836", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @Article{Borwein:2004:RAFb, author = "J. Borwein and R. Crandall", title = "On the {Ramanujan} {AGM} fraction. {II}. {The} complex-parameter case", journal = j-EXP-MATH, volume = "13", number = "3", pages = "287--295", month = "????", year = "2004", CODEN = "????", DOI = "https://doi.org/10.1080/10586458.2004.10504541", ISSN = "1058-6458 (print), 1944-950X (electronic)", ISSN-L = "1058-6458", MRclass = "11J70 (11A55 33C05)", MRnumber = "2103327 (2005h:11149)", MRreviewer = "James G. Mc Laughlin", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "http://projecteuclid.org/euclid.em; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib", URL = "http://docserver.carma.newcastle.edu.au/29/; http://projecteuclid.org/euclid.em/1103749837", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @Article{Bullen:2004:GAM, author = "P. S. Bullen", title = "The geometric--arithmetic mean inequality", journal = "J. Indones. Math. Soc.", volume = "10", number = "2", pages = "99--102", year = "2004", ISSN = "0854-1388", MRclass = "26D15", MRnumber = "2097093", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the Indonesian Mathematical Society. Majalah Ilmiah Himpunan Matematika Indonesia (MIHMI)", } @InCollection{Cox:2004:AGM, author = "David A. Cox", title = "The Arithmetic--Geometric Mean of {Gauss}", crossref = "Berggren:2004:PSB", pages = "481--536", year = "2004", DOI = "https://doi.org/10.1007/978-1-4757-4217-6_55", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_55", acknowledgement = ack-nhfb, } @Article{Ovesea-Tudor:2004:UCC, author = "Horiana Ovesea-Tudor", title = "Univalence criteria connected with arithmetic and geometric means", journal = "Studia Univ. Babe{\c{s}}-Bolyai Math.", volume = "49", number = "1", pages = "55--62", year = "2004", ISSN = "0252-1938", MRclass = "30C55", MRnumber = "2140514", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Universitatis Babe{\c{s}}-Bolyai. Studia. Mathematica", } @InCollection{Salamin:2004:CUA, author = "Eugene Salamin", title = "Computation of $ \pi $ Using Arithmetic--Geometric Mean", crossref = "Berggren:2004:PSB", pages = "418--423", year = "2004", DOI = "https://doi.org/10.1007/978-1-4757-4217-6_46", bibdate = "Tue Mar 14 11:58:19 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_46", acknowledgement = ack-nhfb, } @Article{Mihesan:2005:RPT, author = "Vasile Mihe{\c{s}}an", title = "{Rado} and {Popoviciu} type inequalities for pseudo arithmetic and geometric means", journal = j-INT-J-PURE-APPL-MATH, volume = "23", number = "3", pages = "293--297", year = "2005", ISSN = "1311-8080 (print), 1314-3395 (electronic)", ISSN-L = "1314-3395", MRclass = "26D15 (26E60)", MRnumber = "2176202", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Pure and Applied Mathematics", journal-URL = "http://ijpam.eu/", } @Article{Wu:2005:SRE, author = "Shanhe Wu", title = "Some results on extending and sharpening the {Weierstrass} product inequalities", journal = j-J-MATH-ANAL-APPL, volume = "308", number = "2", pages = "689--702", year = "2005", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/j.jmaa.2004.11.064", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X04010157", abstract = "In this paper, we establish two extensions of Weierstrass's inequality involving symmetric functions by means of the theory of majorization, and give an interesting sharpness of Weierstrass's inequality by using the arithmetic--geometric mean inequality. Furthermore, we apply these results to improve a well-known inequality and deduce some new inequalities.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "Arithmetic--geometric mean inequality; Elementary symmetric function; Majorization; Schur-concave function; Weierstrass's inequality", } @Article{Bhatia:2006:IAG, author = "Rajendra Bhatia", title = "Interpolating the arithmetic--geometric mean inequality and its operator version", journal = j-LINEAR-ALGEBRA-APPL, volume = "413", number = "2--3", pages = "355--363", day = "1", month = mar, year = "2006", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2005.03.005", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "47A63 (15A48)", MRnumber = "2198940", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2000.bib", note = "Special Issue on the 11th Conference of the International Linear Algebra Society, Coimbra, 200411th Conference of the International Linear Algebra Society, Coimbra, 2004", URL = "http://www.sciencedirect.com/science/article/pii/S0024379505001382", abstract = "Two families of means (called Heinz means and Heron means) that interpolate between the geometric and the arithmetic mean are considered. Comparison inequalities between them are established. Operator versions of these inequalities are obtained. Failure of such extensions in some cases is illustrated by a simple example.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Inequalities for means; Operator inequalities; Positive definite matrix; Unitarily invariant norm", } @TechReport{Brent:2006:FAH, author = "Richard P. Brent", title = "Fast Algorithms for High-Precision Computation of Elementary Functions", type = "Report", number = "??", institution = "Australian National University", address = "Canberra, ACT 0200, Australia", pages = "61", day = "12", month = jul, year = "2006", bibdate = "Fri Sep 04 16:33:10 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "http://rnc7.loria.fr/brent_invited.pdf; https://maths-people.anu.edu.au/~brent/pd/RNC7t.pdf", acknowledgement = ack-nhfb, keywords = "arithmetic-geometric mean", remark = "From page 57: ``This talk is based on a chapter of a book that Paul Zimmermann and I are writing''. That book is entry \cite{Brent:2011:MCA}.", } @Article{Enge:2006:CCP, author = "Andreas Enge", title = "The complexity of class polynomial computation via floating point approximations", journal = "arXiv.org", volume = "??", number = "??", pages = "??--??", day = "24", month = jan, year = "2006", CODEN = "????", ISSN = "????", ISSN-L = "????", bibdate = "Wed Sep 30 12:43:49 MDT 2015", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", note = "Published in Mathematics of Computation 78, {\bf 266} (2009) 1089--1107.", URL = "http://arxiv.org/abs/cs/0601104", abstract = "We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic--geometric mean. It runs in time $ O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon }) = O (h^{2 + \epsilon }) $ for any $ \epsilon > 0 $, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $ \log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials.", acknowledgement = ack-nhfb, subject = "Numerical Analysis (cs.NA); Symbolic Computation (cs.SC); Number Theory (math.NT)", } @Article{Holland:2006:IBC, author = "Finbarr Holland", title = "An inequality between compositions of weighted arithmetic and geometric means", journal = j-JIPAM-J-INEQUAL-PURE-APPL-MATH, volume = "7", number = "5", pages = "Article 159, 8", year = "2006", ISSN = "1443-5756", MRclass = "26D15 (26E60)", MRnumber = "2268614", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "JIPAM. Journal of Inequalities in Pure and Applied Mathematics", journal-URL = "http://www.emis.de/journals/JIPAM/", } @Article{Liu:2006:OTA, author = "Lei Liu and Jian Hua Zhang", title = "An operator-trace arithmetic--geometric mean inequality", journal = "J. Baoji Univ. Arts Sci. Math. Colloq. Chin. Univ.", volume = "3B", number = "3B", pages = "208--209", year = "2006", ISSN = "1007-1261", MRclass = "47A63", MRnumber = "2268274", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Baoji University of Arts and Sciences. Math Colloquium of Chinese Universities. Baoji Wenli Xueyuan Xuebao. Daxue Shuxue Jikan", } @Article{Sabnis:2006:AAG, author = "S. V. Sabnis and G. Agnihothram", title = "Application of arithmetic--geometric mean inequality for construction of reliability test plan for parallel systems in the presence of covariates", journal = j-ECON-QUAL-CONTROL, volume = "21", number = "2", pages = "219--230", year = "2006", DOI = "https://doi.org/10.1515/EQC.2006.219", ISSN = "0940-5151 (print), 1869-6147 (electronic)", MRclass = "62N05 (62N03)", MRnumber = "2364112", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Economic Quality Control", journal-URL = "http://degruyter.com/eqc", } @Article{Taneja:2006:GAG, author = "Inder Jeet Taneja", title = "Generalized arithmetic and geometric mean divergence measure and their statistical aspects", journal = "J. Interdiscip. Math.", volume = "9", number = "2", pages = "249--266", year = "2006", DOI = "https://doi.org/10.1080/09720502.2006.10700442", ISSN = "0972-0502", MRclass = "62B10", MRnumber = "2245159", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Interdisciplinary Mathematics", } @Article{Tao:2006:MRS, author = "Yunxing Tao", title = "More results on singular value inequalities of matrices", journal = j-LINEAR-ALGEBRA-APPL, volume = "416", number = "2--3", pages = "724--729", year = "2006", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2005.12.017", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379506000024", abstract = "The arithmetic--geometric mean inequality for singular values due to Bhatia and Kittaneh says that $ 2 s_j (A B^\star) \leq s_j (A^\star A + B^\star B) $, $ j = 1, 2, \ldots $ for any matrices $A$, $B$. We give a new equivalent form and some relevant generalizations of this inequality. In particular, we show that $ s_j (A^{1 / 4} B^{3 / 4} + A^{3 / 4} B^{1 / 4}) \leq s_j (A + B)$, $ j = 1, \ldots, n$ for any $ n \times n $ positive semidefinite matrices $A$, $B$, which proves a special case of Zhan's conjecture posed in 2000.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Arithmetic--geometric mean; Positive semidefinite matrix; Singular value", } @Article{Yamazaki:2006:EKI, author = "Takeaki Yamazaki", title = "An extension of {Kantorovich} inequality to $n$-operators via the geometric mean by {Ando--Li--Mathias}", journal = j-LINEAR-ALGEBRA-APPL, volume = "416", number = "2--3", pages = "688--695", year = "2006", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2005.12.013", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S002437950500604X", abstract = "In this paper, we shall extend Kantorovich inequality. This is an estimate by using the geometric mean of n-operators which have been defined by Ando--Li--Mathias in [T. Ando, C. K. Li, R. Mathias, Geometric means, Linear Algebra Appl. 385 (2004) 305--334]. As a related result, we obtain a converse of arithmetic--geometric means inequality of n-operators via Kantorovich constant.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Arithmetic--geometric means inequality; Geometric mean of n-operators; Kantorovich inequality; Specht's ratio", } @Article{Aujla:2007:EIC, author = "Jaspal Singh Aujla and Jean-Christophe Bourin", title = "Eigenvalue inequalities for convex and log-convex functions", journal = j-LINEAR-ALGEBRA-APPL, volume = "424", number = "1", pages = "25--35", year = "2007", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2006.02.027", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "Special Issue in honor of Roger Horn", URL = "http://www.sciencedirect.com/science/article/pii/S0024379506001261", abstract = "We give a matrix version of the scalar inequality $ f(a + b) \leq f(a) + f(b) $ for positive concave functions $f$ on $ [0, \infty)$. We show that Choi's inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic--geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia--Kittaneh arithmetic--geometric mean inequality.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Convex function; Eigenvalue; Majorization; Unital positive linear map", } @Article{Barnard:2007:IHA, author = "Roger W. Barnard and Kendall C. Richards", title = "On inequalities for hypergeometric analogues of the arithmetic--geometric mean", journal = j-JIPAM-J-INEQUAL-PURE-APPL-MATH, volume = "8", number = "3", pages = "Article 65, 5", year = "2007", ISSN = "1443-5756", MRclass = "26D15 (26E60 33C05)", MRnumber = "2345920", MRreviewer = "Giampietro Allasia", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "JIPAM. Journal of Inequalities in Pure and Applied Mathematics", journal-URL = "http://www.emis.de/journals/JIPAM/", } @Article{Hirschhorn:2007:GI, author = "M. D. Hirschhorn", title = "The {AM--GM} inequality", journal = j-MATH-INTEL, volume = "29", number = "4", pages = "7--??", month = "????", year = "2007", CODEN = "MAINDC", DOI = "https://doi.org/10.1007/BF02986168", ISSN = "0343-6993 (print), 1866-7414 (electronic)", ISSN-L = "0343-6993", bibdate = "Fri Feb 15 16:15:39 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", ZMnumber = "1225.00022", acknowledgement = ack-nhfb, fjournal = "The Mathematical Intelligencer", journal-URL = "http://link.springer.com/journal/283", ZMclass = "00A35 (Methodology of mathematics, didactics); 26D15 (Inequalities for sums, series and integrals of real functions); 97I20 (Mappings and functions (educational aspects))", } @Article{Kim:2007:CIH, author = "Sejong Kim and Yongdo Lim", title = "A converse inequality of higher order weighted arithmetic and geometric means of positive definite operators", journal = j-LINEAR-ALGEBRA-APPL, volume = "426", number = "2--3", pages = "490--496", day = "15", month = oct, year = "2007", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2007.05.028", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "47A63", MRnumber = "2350672", MRreviewer = "Chia-Shiang Lin", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379507002455", abstract = "In this paper we consider weighted arithmetic and geometric means of higher orders constructed by the symmetrization method appeared in Ando--Li--Mathias's definition of multi-variable geometric means and the arithmetic--geometric mean inequality of higher order weighted version. We establish a converse inequality of higher order weighed arithmetic and geometric means via Specht ratio.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Arithmetic--geometric mean inequality; Higher order weighted geometric mean; Positive definite operator; Specht ratio", } @Book{King:2007:DNC, author = "Louis Vessot King", title = "On the Direct Numerical Calculation of Elliptic Functions and Integrals", publisher = "Mellon Press", address = "", pages = "56", year = "2007", ISBN = "1-4067-4226-0", ISBN-13 = "978-1-4067-4226-8", LCCN = "", bibdate = "Wed Feb 03 08:53:04 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib", acknowledgement = ack-nhfb, remark = "The AGM method for Jacobian elliptic was discovered by this book's author at McGill University in 1913, first published in \cite{King:1921:SNF}, and then in a 1924 monograph, of which this is a reprint.", } @Article{Koike:2007:IFP, author = "Kenji Koike and Hironori Shiga", title = "Isogeny formulas for the {Picard} modular form and a three terms arithmetic geometric mean", journal = j-J-NUMBER-THEORY, volume = "124", number = "1", pages = "123--141", month = may, year = "2007", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2006.08.002", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", MRclass = "11F55", MRnumber = "2320994", MRreviewer = "Enrique Gonz{\'a}lez-Jim{\'e}nez", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X06002058", abstract = "In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are ``Neben type'' modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM.", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Lehavi:2007:EFA, author = "D. Lehavi and C. Ritzenthaler", title = "An Explicit Formula for the Arithmetic--Geometric Mean in Genus $3$", journal = j-EXP-MATH, volume = "16", number = "4", pages = "421--440", month = "????", year = "2007", CODEN = "????", ISSN = "1058-6458 (print), 1944-950x (electronic)", ISSN-L = "1058-6458", MRclass = "14H40 (14H45 14Q05)", MRnumber = "2378484", MRreviewer = "Mich{\`e}le Pelletier", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://projecteuclid.org/euclid.em; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib; http://www.tandfonline.com/toc/uexm20/16/4", URL = "http://projecteuclid.org/euclid.em/1204836513", acknowledgement = ack-nhfb, fjournal = "Experimental Mathematics", journal-URL = "http://www.tandfonline.com/loi/uexm20", } @Article{Tanimoto:2007:NOA, author = "Shinji Tanimoto", title = "A novel operation associated with {Gauss}' arithmetic--geometric means", journal = "arXiv.org", volume = "??", number = "??", pages = "1--6", day = "27", month = aug, year = "2007", bibdate = "Tue Mar 14 18:08:54 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "https://arxiv.org/pdf/0708.3521.pdf", abstract = "The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic--geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic properties are proved.", acknowledgement = ack-nhfb, } @Article{Walden:2007:HMI, author = "Byron L. Walden and Lesley A. Ward", title = "A harmonic measure interpretation of the arithmetic--geometric mean", journal = j-AMER-MATH-MONTHLY, volume = "114", number = "7", pages = "610--622", year = "2007", CODEN = "AMMYAE", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", MRclass = "31A15 (26E60 28A12 30C20 30C85 33C45)", MRnumber = "2341324", MRreviewer = "Dimitrios Betsakos", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Aldaz:2008:RIB, author = "J. M. Aldaz", title = "A refinement of the inequality between arithmetic and geometric means", journal = j-J-MATH-INEQUAL, volume = "2", number = "4", pages = "473--477", year = "2008", DOI = "https://doi.org/10.7153/jmi-02-42", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26D15 (26E60)", MRnumber = "2482460", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Bhatia:2008:MAG, author = "Rajendra Bhatia and Fuad Kittaneh", title = "The matrix arithmetic--geometric mean inequality revisited", journal = j-LINEAR-ALGEBRA-APPL, volume = "428", number = "8--9", pages = "2177--2191", day = "15", month = apr, year = "2008", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2007.11.030", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A42 (47A30 47A63)", MRnumber = "2401646", MRreviewer = "Omar Hirzallah", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379507005381", abstract = "Ideas related to matrix versions of the arithmetic--geometric mean inequality are explained.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Matrix inequalities; Matrix monotone functions; Pinching; Singular values; Unitarily invariant norms", } @Article{Carvalhaes:2008:APS, author = "Claudio G. Carvalhaes and Patrick Suppes", title = "Approximations for the period of the simple pendulum based on the arithmetic--geometric mean", journal = j-AMER-J-PHYSICS, volume = "76", number = "12", pages = "1150--1154", month = dec, year = "2008", CODEN = "AJPIAS", DOI = "https://doi.org/10.1119/1.2968864", ISSN = "0002-9505 (print), 1943-2909 (electronic)", bibdate = "Tue Mar 14 18:22:35 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", note = "See comments in \cite{Villarino:2014:ASP} about prior work before 1966 by Albert Edward Ingham (1900--1967) producing both upper and lower bounds to approximations to the period of a pendulum.", URL = "http://aapt.scitation.org/doi/full/10.1119/1.2968864", acknowledgement = ack-nhfb, fjournal = "American Journal of Physics", journal-URL = "http://scitation.aip.org/content/aapt/journal/ajp", remark = "From the introduction: ``The lack of an elementary closed-form restricts the study of the simple pendulum at the undergraduate level to small oscillations. \ldots{} convergence with eight digits of accuracy is obtained for an amplitude of $ 179^\circ $ after only four iterations.''", } @Article{Feng:2008:MVS, author = "Bao Qi Feng and Andrew Tonge", title = "Matrix versions of some refinements of the arithmetic--geometric mean inequality", journal = j-J-MATH-SCI-ADV-APPL, volume = "1", number = "2", pages = "243--264", year = "2008", ISSN = "0974-5750", MRclass = "26E60 (15A45)", MRnumber = "2530506", MRreviewer = "Alan L. Horwitz", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Sciences. Advances and Applications", journal-URL = "http://www.scientificadvances.co.in/about-this-journal/1", } @Article{Ito:2008:MAG, author = "Takashi Ito", title = "Mixed arithmetic and geometric means and related inequalities", journal = j-JIPAM-J-INEQUAL-PURE-APPL-MATH, volume = "9", number = "3", pages = "Article 65, 21", year = "2008", ISSN = "1443-5756", MRclass = "26D20 (26E60)", MRnumber = "2476645", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "JIPAM. Journal of Inequalities in Pure and Applied Mathematics", journal-URL = "http://www.emis.de/journals/JIPAM/", } @Article{Jarvis:2008:HGA, author = "Frazer Jarvis", title = "Higher genus arithmetic--geometric means", journal = j-RAMANUJAN-J, volume = "17", number = "1", pages = "1--17", year = "2008", CODEN = "RAJOF9", DOI = "https://doi.org/10.1007/s11139-007-9058-0", ISSN = "1382-4090 (print), 1572-9303 (electronic)", ISSN-L = "1382-4090", MRclass = "14K25 (11B83 11F27)", MRnumber = "2439522", MRreviewer = "Cristiana Bertolin", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan", journal-URL = "http://link.springer.com/journal/11139", } @Article{Koike:2008:EGA, author = "Kenji Koike and Hironori Shiga", title = "An extended {Gauss} {AGM} and corresponding {Picard} modular forms", journal = j-J-NUMBER-THEORY, volume = "128", number = "7", pages = "2097--2126", month = jul, year = "2008", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2007.12.001", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Tue Mar 14 17:07:32 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X08000218", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", } @Article{Lorentzen:2008:CDR, author = "Lisa Lorentzen", title = "Convergence and divergence of the {Ramanujan} {AGM} fraction", journal = j-RAMANUJAN-J, volume = "16", number = "1", pages = "83--95", month = may, year = "2008", DOI = "https://doi.org/10.1007/s11139-007-9112-y", ISSN = "1382-4090 (print), 1572-9303 (electronic)", ISSN-L = "1382-4090", bibdate = "Tue Mar 14 15:27:34 2017", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", abstract = "We prove that the Ramanujan AGM fraction diverges if $ |a| = |b| $ with $ a^2 \neq b^2 $. Thereby we prove two conjectures posed by J. Borwein and R. Crandall. We also demonstrate a method for accelerating the convergence of this continued fraction when it converges.", acknowledgement = ack-nhfb, fjournal = "The {Ramanujan} Journal", journal-URL = "http://link.springer.com/journal/11139", } @Article{Micic:2008:IIA, author = "Jadranka Mi{\'c}i{\'c} and Josip Pe{\v{c}}ari{\'c} and Vidosava {\v{S}}imi{\'c}", title = "Inequalities involving the arithmetic and geometric operator means", journal = j-MATH-INEQUAL-APPL, volume = "11", number = "3", pages = "415--430", year = "2008", DOI = "https://doi.org/10.7153/mia-11-31", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "47A63", MRnumber = "2431206", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Uchida:2008:SPG, author = "Yasuharu Uchida", title = "A simple proof of the geometric--arithmetic mean inequality", journal = j-JIPAM-J-INEQUAL-PURE-APPL-MATH, volume = "9", number = "2", pages = "Article 56, 2", year = "2008", ISSN = "1443-5756", MRclass = "26E60", MRnumber = "2417338", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "JIPAM. Journal of Inequalities in Pure and Applied Mathematics", journal-URL = "http://www.emis.de/journals/JIPAM/", } @Article{Yeh:2008:SEF, author = "Cheh-Chih Yeh and Hung-Wen Yeh and Wenyaw Chan", title = "Some equivalent forms of the arithmetic--geometric mean inequality in probability: a survey", journal = j-J-INEQUAL-APPL, pages = "Art. ID 386715, 9", year = "2008", ISSN = "1025-5834", ISSN-L = "1025-5834", MRclass = "26E60 (60E15)", MRnumber = "2449062", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Aldaz:2009:SII, author = "J. M. Aldaz", title = "Self-improvement of the inequality between arithmetic and geometric means", journal = j-J-MATH-INEQUAL, volume = "3", number = "2", pages = "213--216", year = "2009", DOI = "https://doi.org/10.7153/jmi-03-21", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26D15", MRnumber = "2542299", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Cheon:2009:RBS, author = "Gi-Sang Cheon and Andrew W. Eckford", title = "A relationship between subpermanents and the arithmetic--geometric mean inequality", journal = j-LINEAR-ALGEBRA-APPL, volume = "430", number = "1", pages = "114--120", day = "1", month = jan, year = "2009", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2008.07.001", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A15 (15A48)", MRnumber = "2460503", MRreviewer = "Carlos M. da Fonseca", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2000.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379508003406", abstract = "Using the arithmetic--geometric mean inequality, we give bounds for k-subpermanents of nonnegative $ n \times n $ matrices $F$. In the case $ k = n$, we exhibit an $ n^2$-set $S$ whose arithmetic and geometric means constitute upper and lower bounds for $ \per (F) / n!$. We offer sharpened versions of these bounds when $F$ has zero-valued entries.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "AM--GM inequality; Permanent; Subpermanent", } @Article{Furuichi:2009:TIP, author = "Shigeru Furuichi and Ken Kuriyama and Kenjiro Yanagi", title = "Trace inequalities for products of matrices", journal = j-LINEAR-ALGEBRA-APPL, volume = "430", number = "8--9", pages = "2271--2276", year = "2009", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2008.12.003", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379508005685", abstract = "In this short paper, we study some trace inequalities of the products of the matrices and the power of matrices by the use of elementary calculations.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Arithmetic--geometric mean inequality and nonnegative matrix; Matrix trace inequalitiy", } @Article{Hayashi:2009:NCA, author = "Tomohiro Hayashi", title = "Non-commutative arithmetic--geometric mean inequality", journal = j-PROC-AM-MATH-SOC, volume = "137", number = "10", pages = "3399--3406", year = "2009", CODEN = "PAMYAR", DOI = "https://doi.org/10.1090/S0002-9939-09-09911-0", ISSN = "0002-9939 (print), 1088-6826 (electronic)", ISSN-L = "0002-9939", MRclass = "47A63", MRnumber = "2515409", MRreviewer = "{\`E}dward L. Pekarev", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Proceedings of the American Mathematical Society", journal-URL = "http://www.ams.org/journals/proc", } @Article{Kim:2009:SCI, author = "Sejong Kim and Hosoo Lee and Yongdo Lim", title = "A sharp converse inequality of three weighted arithmetic and geometric means of positive definite operators", journal = j-MATH-INEQUAL-APPL, volume = "12", number = "3", pages = "519--523", year = "2009", DOI = "https://doi.org/10.7153/mia-12-40", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "47A63", MRnumber = "2540975", MRreviewer = "Pedro Tradacete", bibdate = "Tue Aug 15 10:24:29 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Kosheleva:2009:GNJ, author = "O. Kosheleva and V. Kreinovich", title = "Guesstimation: a New Justification of the Geometric Mean Heuristic", journal = j-APPL-MATH-SCI-RUSE, volume = "3", number = "47", pages = "2335--2342", year = "2009", ISSN = "1312-885x (print), 1314-7552 (electronic)", MRclass = "62F10", MRnumber = "MR2558236 (2010i:62055)", bibdate = "Fri Oct 15 09:08:58 2010", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.cs.utep.edu/vladik/2009/tr09-10.pdf; http://www.openj-gate.com/Browse/ArticleList.aspx?Journal_id=124136&issue_id=1153557", abstract = "In many practical situations in which the only information we have about the quantity $x$ is that its value is within an interval $ [\underbar {x}; \overbar {x}]$, a responsible estimate for this quantity is the geometric mean of the bounds $ \sqrt {\underbar {x} \cdot \overbar {x}}$. In this paper, we provide a new justification for this geometric mean heuristic.", acknowledgement = ack-nhfb, fjournal = "Applied Mathematical Sciences (Ruse)", journal-URL = "http://www.m-hikari.com/ams/", keywords = "arithmetic mean; geometric mean; interval arithmetic", } @Article{Lekner:2009:ASC, author = "John Lekner", title = "Axially symmetric charge distributions and the arithmetic--geometric mean", journal = j-J-ELECTROST, volume = "67", number = "6", pages = "880--885", year = "2009", CODEN = "JOELDH", DOI = "https://doi.org/10.1016/j.elstat.2009.07.007", ISSN = "0304-3886 (print), 1873-5738 (electronic)", ISSN-L = "0304-3886", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0304388609001806", abstract = "The potential at an arbitrary point in space due to an axially symmetric charge distribution is related to the arithmetic--geometric mean of the maximum and minimum distances from each annulus of constant charge density. The arithmetic--geometric mean is expressible in terms of the elliptic integral of the first kind, $K$. Thus the potential of a charged body with cylindrical symmetry is reducible to a double integral over the charge density times $K$. For conductors the charge resides on the surface, and the potential reduces to a single integral over the surface charge density times $K$. This result leads to a new proof of the relation between a sum over products of Legendre polynomials and the complete elliptic integral of the first kind, and to new identities for the angular average of Legendre polynomials divided by $ | r - r' |$. The method also provides a direct route to the capacitance of a slender torus, without the use of toroidal coordinates.", acknowledgement = ack-nhfb, ajournal = "J. Electrost.", fjournal = "Journal of Electrostatics", journal-URL = "http://www.sciencedirect.com/science/journal/03043886", keywords = "Arithmetic--geometric mean; Axially symmetric charge distributions; Thin torus", } @Article{Qi:2009:AUP, author = "Feng Qi and Anthony Sofo", title = "An alternative and united proof of a double inequality for bounding the arithmetic--geometric mean", journal = "Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys.", volume = "71", number = "3", pages = "69--76", year = "2009", ISSN = "1223-7027", MRclass = "26E60 (26D15)", MRnumber = "2553929", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "``Politehnica'' University of Bucharest. Scientific Bulletin. Series A. Applied Mathematics and Physics", } @Article{Raissouli:2009:AGH, author = "Mustapha Ra{\"\i}ssouli and Fatima Leazizi and Mohamed Chergui", title = "Arithmetic--geometric--harmonic mean of three positive operators", journal = j-JIPAM-J-INEQUAL-PURE-APPL-MATH, volume = "10", number = "4", pages = "Article 117, 11", year = "2009", ISSN = "1443-5756", MRclass = "15B48 (47A64)", MRnumber = "2577887", MRreviewer = "Mohammad Sal Moslehian", bibdate = "Tue Aug 15 10:18:13 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "JIPAM. Journal of Inequalities in Pure and Applied Mathematics", journal-URL = "http://www.emis.de/journals/JIPAM/", } @Article{Aldaz:2010:CRB, author = "J. M. Aldaz", title = "Concentration of the ratio between the geometric and arithmetic means", journal = j-J-THEOR-PROBAB, volume = "23", number = "2", pages = "498--508", year = "2010", CODEN = "JTPREO", DOI = "https://doi.org/10.1007/s10959-009-0215-9", ISSN = "0894-9840 (print), 1572-9230 (electronic)", ISSN-L = "0894-9840", MRclass = "26E60 (26D15 28A75 60D05)", MRnumber = "2644872", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Theoretical Probability", journal-URL = "http://link.springer.com/journal/10959", } @InCollection{Arndt:2010:AEI, author = "J{\"o}rg Arndt", title = "The {AGM}, elliptic integrals, and algorithms for computing $ \pi $", crossref = "Arndt:2011:MC", publisher = pub-SV, address = pub-SV:adr, pages = "599--621", year = "2011", DOI = "https://doi.org/10.1007/978-3-642-14764-7_31", bibdate = "Tue Mar 14 15:06:12 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Unpublished{Brent:2010:MPZ, author = "Richard P. Brent", title = "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", day = "20", month = apr, year = "2010", MRclass = "11Y60 (Primary), 65Y20 (Secondary)", bibdate = "Tue Apr 26 14:13:36 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", note = "Reprint of \cite{Brent:1976:MPZ} with a postscript describing more recent developments. See also \cite{Salamin:1976:CUA}", URL = "http://arxiv.org/abs/1004.3412v2; http://wwwmaths.anu.edu.au/~brent/pub/pub028.html", abstract = "We consider methods for finding high-precision approximations to simple zeros of smooth functions. As an application, we give fast methods for evaluating the elementary functions $ \log (x) $, $ \exp (x) $, $ \sin (x) $ etc. to high precision. For example, if $x$ is a positive floating-point number with an $n$-bit fraction, then (under rather weak assumptions) an $n$-bit approximation to $ \log (x)$ or $ \exp (x)$ may be computed in time asymptotically equal to $ 13 M(n) \lg (n)$, where $ M(n)$ is the time required to multiply floating-point numbers with $n$-bit fractions. Similar results are given for the other elementary functions. Some analogies with operations on formal power series (over a field of characteristic zero) are discussed. In particular, it is possible to compute the first $n$ terms in $ \log (1 + a_1 x + \cdots)$ or $ \exp (a_1. x) + \cdots $ in time $ O(M(n))$, where $ M(n)$ is the time required to multiply two polynomials of degree $ n - 1$. It follows that the first $n$ terms in a $q$-th power $ (1 + a_1 x + \cdots)^q$ can be computed in time $ O(M(n))$, independent of $q$. One of the results of this paper is the ``Gauss--Legendre'' or ``Brent--Salamin'' algorithm for computing pi. This is the first quadratically convergent algorithm for pi. It was also published in Brent [J. ACM 23 (1976), 242--251], and independently by Salamin [Math. Comp. 30 (1976), 565--570].", acknowledgement = ack-nhfb, } @Article{Brent:2010:UAE, author = "Richard P. Brent", title = "Unrestricted algorithms for elementary and special functions", journal = "arXiv.org", volume = "??", number = "??", pages = "1--13", month = apr, year = "2010", bibdate = "Sat Feb 25 10:56:45 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://arxiv.org/abs/1004.3621", abstract = "We describe some ``unrestricted'' algorithms which are useful for the computation of elementary and special functions when the precision required is not known in advance. Several general classes of algorithms are identified and illustrated by examples. The topics include: power series methods, use of halving identities, asymptotic expansions, continued fractions, recurrence relations, Newton's method, numerical contour integration, and the arithmetic--geometric mean. Most of the algorithms discussed are implemented in the MP package.", acknowledgement = ack-nhfb, } @Article{Cardenas-Barron:2010:EMD, author = "Leopoldo Eduardo C{\'a}rdenas-Barr{\'o}n", title = "An easy method to derive {EOQ} and {EPQ} inventory models with backorders", journal = j-COMPUT-MATH-APPL, volume = "59", number = "2", pages = "948--952", year = "2010", CODEN = "CMAPDK", DOI = "https://doi.org/10.1016/j.camwa.2009.09.013", ISSN = "0898-1221 (print), 1873-7668 (electronic)", ISSN-L = "0898-1221", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0898122109006774", abstract = "Recently, a cost minimization method to determine the lot size for the EOQ/EPQ models with backorders was published. This method is based on the well-known arithmetic--geometric mean inequality. Although the cost minimization method is correct and interesting, it does not focus on deriving the backorders level. This paper proposes another simple approach. The proposed method finds both the lot size and the backorders level.", acknowledgement = ack-nhfb, fjournal = "Computers and Mathematics with Applications", journal-URL = "http://www.sciencedirect.com/science/journal/08981221", keywords = "Algebraic optimization; Arithmetic--geometric mean (AGM) inequality; Backorders level; Cauchy--Bunyakovsky--Schwarz (CBS) inequality; Cost comparisons optimization; Economic order quantity; Economic production quantity", } @Article{Cardenas-Barron:2010:SMC, author = "Leopoldo Eduardo C{\'a}rdenas-Barr{\'o}n", title = "A simple method to compute economic order quantities: Some observations", journal = j-APPL-MATH-MODEL, volume = "34", number = "6", pages = "1684--1688", year = "2010", CODEN = "AMMODL", DOI = "https://doi.org/10.1016/j.apm.2009.08.024", ISSN = "0307-904x (print), 1872-8480 (electronic)", ISSN-L = "0307-904X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0307904X09002777", abstract = "Teng [2] presents an arithmetic--geometric mean method to be applied to determine the optimal lot size for the EOQ/EPQ models, taking into account backorders. Although the arithmetic--geometric mean method is correct, arguments as to when (not) to use the arithmetic--geometric mean inequality as optimization method are not complete. Moreover, this optimization method does not focus on the method for deriving the optimal backorders level. The main purpose of this work is to overcome these shortcomings, presents a discussion of when (not) to use the cost minimization method and derives the optimal backorders level.", acknowledgement = ack-nhfb, fjournal = "Applied mathematical modelling", keywords = "Algebraic optimization; Arithmetic--geometric mean optimization; Backorders; Cost comparisons optimization; Cost-difference comparisons optimization; EOQ/EPQ", } @Article{Kinjo:2010:AAG, author = "Kensaku Kinjo and Yuken Miyasaka", title = "$2$-{Adic} arithmetic--geometric mean and elliptic curves", journal = j-INTERDISCIP-INFORM-SCI, volume = "16", number = "1", pages = "5--15", year = "2010", DOI = "https://doi.org/10.4036/iis.2010.5", ISSN = "1340-9050 (print), 1347-6157 (electronic)", ISSN-L = "1340-9050", MRclass = "11S85 (11G07)", MRnumber = "2648110", MRreviewer = "Maria Sabitova", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Interdisciplinary Information Sciences", journal-URL = "https://www.jstage.jst.go.jp/browse/iis", } @Article{Long:2010:OIG, author = "Bo-Yong Long and Yu-Ming Chu", title = "Optimal inequalities for generalized logarithmic, arithmetic, and geometric means", journal = j-J-INEQUAL-APPL, pages = "806825:1--806825:10", year = "2010", DOI = "https://doi.org/10.1155/2010/806825", ISSN = "1025-5834", ISSN-L = "1025-5834", MRclass = "26E60 (26D15)", MRnumber = "2600201", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Maksa:2010:ETF, author = "Gyula Maksa and Adrienn Varga", title = "The equivalence of two functional equations involving the arithmetic mean, the geometric mean and their {Gauss} composition", journal = j-AEQUATIONES-MATHEMATICAE, volume = "80", number = "1-2", pages = "173--179", year = "2010", CODEN = "AEMABN", DOI = "https://doi.org/10.1007/s00010-010-0030-5", ISSN = "0001-9054 (print), 1420-8903 (electronic)", ISSN-L = "0001-9054", MRclass = "39B22 (26E60)", MRnumber = "2736948", MRreviewer = "Tomasz Szostok", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Aequationes Mathematicae", journal-URL = "http://link.springer.com/journal/10", } @Article{Matejicka:2010:POO, author = "Ladislav Matej{\'\i}{\v{c}}ka", title = "Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means", journal = j-J-INEQUAL-APPL, pages = "902432:1--902432:5", year = "2010", DOI = "https://doi.org/10.1155/2010/902432", ISSN = "1025-5834", ISSN-L = "1025-5834", MRclass = "26E60 (26D20)", MRnumber = "2738681", MRreviewer = "Yu-ming Chu", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Matsumoto:2010:AGM, author = "Keiji Matsumoto and Tomohide Terasoma", title = "Arithmetic--geometric means for hyperelliptic curves and {Calabi--Yau} varieties", journal = j-INT-J-MATH, volume = "21", number = "7", pages = "939--949", year = "2010", DOI = "https://doi.org/10.1142/S0129167X1000632X", ISSN = "0129-167X", MRclass = "14K20 (14J32 14K25)", MRnumber = "2671531", MRreviewer = "Ahmed Lesfari", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematics", journal-URL = "http://www.worldscientific.com/worldscinet/ijm", } @Article{Solak:2010:NEN, author = "S{\"u}leyman Solak and Mine Aytekin", title = "A note on the {Euclidean} norms of matrices with arithmetic--geometric harmonic means", journal = j-APPL-MATH-SCI-RUSE, volume = "4", number = "29-32", pages = "1553--1561", year = "2010", ISSN = "1312-885x (print), 1314-7552 (electronic)", MRclass = "15A60", MRnumber = "2643782", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematical Sciences", journal-URL = "http://www.m-hikari.com/ams/", } @Article{Yamamoto:2010:HGA, author = "Kouji Yamamoto and Nobuko Miyamoto and Sadao Tomizawa", title = "Harmonic, geometric and arithmetic means type uncertainty measures for two-way contingency tables with nominal categories", journal = j-ADV-APPL-STAT, volume = "17", number = "2", pages = "143--159", month = aug, year = "2010", CODEN = "????", ISSN = "0972-3617", ISSN-L = "0972-3617", MRclass = "62H17", MRnumber = "2789363", bibdate = "Wed Aug 16 09:05:25 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/advapplstat.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.pphmj.com/abstract/5234.htm", acknowledgement = ack-nhfb, fjournal = "Advances and Applications in Statistics", journal-URL = "http://www.pphmj.com/journals/contents/adas.htm", } @Article{Aldaz:2011:CDB, author = "J. M. Aldaz", title = "Comparison of differences between arithmetic and geometric means", journal = j-TAMKANG-J-MATH, volume = "42", number = "4", pages = "453--462", year = "2011", DOI = "https://doi.org/10.5556/j.tkjm.42.2011.453-462", ISSN = "0049-2930 (print), 2073-9826 (electronic)", ISSN-L = "2073-9826", MRclass = "26E60 (26D15)", MRnumber = "2862349", MRreviewer = "Alfred Witkowski", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Tamkang Journal of Mathematics", journal-URL = "http://journals.math.tku.edu.tw/index.php/TKJM", } @Article{Bayat:2011:AGM, author = "M. Bayat and H. Teimoori", title = "{Arithmetic--Geometric Mean} determinantal identity", journal = j-LINEAR-ALGEBRA-APPL, volume = "435", number = "11", pages = "2936--2941", day = "1", month = dec, year = "2011", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2011.05.031", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A24 (15A15)", MRnumber = "2825293", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S002437951100440X", abstract = "In this paper, we give a generalization of a determinantal identity posed by Charles R. Johnson about minors of a Toeplitz matrix satisfying a specific matrix identity. These minors are those appear in the Dodgson's condensation formula.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795/", keywords = "AGM (arithmetic--geometric mean); Arithmetic--Geometric Mean; Determinantal identity; Dodgson's condensation; Toeplitz matrix", } @Article{Bini:2011:NCM, author = "Dario Andrea Bini and Bruno Iannazzo", title = "A note on computing matrix geometric means", journal = j-ADV-COMPUT-MATH, volume = "35", number = "2--4", pages = "175--192", month = nov, year = "2011", CODEN = "ACMHEX", DOI = "https://doi.org/10.1007/s10444-010-9165-0", ISSN = "1019-7168 (print), 1572-9044 (electronic)", ISSN-L = "1019-7168", MRclass = "65F30 (15A15 15B48)", MRnumber = "2827085", bibdate = "Sat Feb 3 18:22:54 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/advcomputmath.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/article/10.1007/s10444-010-9165-0", acknowledgement = ack-nhfb, fjournal = "Advances in Computational Mathematics", journal-URL = "http://link.springer.com/journal/10444", } @Article{Carls:2011:GTC, author = "Robert Carls", title = "{Galois} Theory of the Canonical Theta Structure", journal = j-INT-J-NUMBER-THEORY, volume = "7", number = "1", pages = "173--202", month = feb, year = "2011", DOI = "https://doi.org/10.1142/S1793042111003934", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", bibdate = "Tue Jul 21 10:01:24 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042111003934", abstract = "In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain $p$-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Casquilho:2011:MDA, author = "Miguel Casquilho and Jorge Buescu", title = "A minimum distance: arithmetic and harmonic means in a geometric dispute", journal = j-INT-J-MATH-EDU-SCI-TECH, volume = "42", number = "3", pages = "399--405", year = "2011", CODEN = "IJMEBM", DOI = "https://doi.org/10.1080/0020739X.2010.526253", ISSN = "0020-739x (print), 1464-5211 (electronic)", ISSN-L = "0020-739X", MRclass = "26E60 (51M16)", MRnumber = "2787160", MRreviewer = "Yu Dong Wu", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Education in Science and Technology", journal-URL = "http://www.tandfonline.com/loi/tmes20", } @Article{Chu:2011:OCC, author = "Yu-Ming Chu and Cheng Zong and Gen-Di Wang", title = "Optimal convex combination bounds of {Seiffert} and geometric means for the arithmetic mean", journal = j-J-MATH-INEQUAL, volume = "5", number = "3", pages = "429--434", year = "2011", DOI = "https://doi.org/10.7153/jmi-05-37", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26E60 (26D07)", MRnumber = "2865559", MRreviewer = "P{\'a}l Burai", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Chu:2011:OIB, author = "Yu-Ming Chu and Miao-Kun Wang", title = "Optimal inequalities between harmonic, geometric, logarithmic, and arithmetic--geometric means", journal = j-J-APPL-MATH, pages = "618929:1--618929:9", year = "2011", DOI = "https://doi.org/10.1155/2011/618929", ISSN = "1110-757x (print), 1687-0042 (electronic)", ISSN-L = "1110-757X", MRclass = "26E60", MRnumber = "2846444", MRreviewer = "Raghib M. Abu-Saris", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Mathematics", journal-URL = "http://www.hindawi.com/journals/jam/", } @Article{Dupont:2011:FEM, author = "R{\'e}gis Dupont", title = "Fast evaluation of modular functions using {Newton} iterations and the {AGM}", journal = j-MATH-COMPUT, volume = "80", number = "275", pages = "1823--1847", month = jul, year = "2011", CODEN = "MCMPAF", DOI = "https://doi.org/10.1090/S0025-5718-2011-01880-6", ISSN = "0025-5718 (print), 1088-6842 (electronic)", ISSN-L = "0025-5718", bibdate = "Mon Apr 18 06:32:30 MDT 2011", bibsource = "http://www.ams.org/mcom/2011-80-275; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib", URL = "http://www.ams.org/journals/mcom/2011-80-275/S0025-5718-2011-01880-6/; http://www.ams.org/journals/mcom/2011-80-275/S0025-5718-2011-01880-6/S0025-5718-2011-01880-6.pdf", acknowledgement = ack-nhfb, fjournal = "Mathematics of Computation", journal-URL = "http://www.ams.org/mcom/", } @Article{Freour:2011:MCS, author = "S. Fr{\'e}our and E. Lacoste and J. Fajoui and F. Jacquemin", title = "On the meaning of the chosen set-averaging method within {Eshelby--Kr{\"o}ner} self-consistent scale transition model: the geometric mean versus the classical arithmetic average", journal = j-Z-ANGE-MATH-MECH, volume = "91", number = "9", pages = "689--698", year = "2011", CODEN = "ZAMMAX", DOI = "https://doi.org/10.1002/zamm.201000167", ISSN = "0044-2267 (print), 1521-4001 (electronic)", ISSN-L = "0044-2267", MRclass = "74Q20", MRnumber = "2838542", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "ZAMM. Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik. Journal of Applied Mathematics and Mechanics", journal-URL = "http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-4001", } @Book{Gauss:2011:W, author = "Carl Friedrich Gauss", title = "Werke", volume = "3", publisher = pub-CAMBRIDGE, address = pub-CAMBRIDGE:adr, pages = "514", year = "2011", DOI = "https://doi.org/10.1017/CBO9781139058247", ISBN = "1-108-03225-7 (paperback), 1-139-05824-X (e-book)", ISBN-13 = "978-1-108-03225-4 (paperback), 978-1-139-05824-7 (e-book)", LCCN = "????", bibdate = "Tue Mar 14 18:59:37 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Cambridge library collection. Mathematics", abstract = "The genius of Carl Friedrich Gauss (1777-1855) and the novelty of his work (published in Latin, German, and occasionally French) in areas as diverse as number theory, probability and astronomy were already widely acknowledged during his lifetime. But it took another three generations of mathematicians to reveal the true extent of his output as they studied Gauss' extensive unpublished papers and his voluminous correspondence. This posthumous twelve-volume collection of Gauss' complete works, published between 1863 and 1933, marks the culmination of their efforts and provides a fascinating account of one of the great scientific minds of the nineteenth century. Volume 3, which appeared in 1866, focuses on analysis. It includes Gauss' work on elliptic functions and on power series, for which he gave the first convergence criteria, as well as his first (1799) proof of the fundamental theorem of algebra, and reviews of works by contemporaries including Fourier.", acknowledgement = ack-nhfb, author-dates = "1777--1855", remark = "Originally published in G{\"o}ttingen: Koniglichen Gesellschaft der Wissenschaften, 1866", } @Article{Gumus:2011:SVI, author = "Ibrahim Halil Gumus and Omar Hirzallah and Necati Taskara", title = "Singular value inequalities for the arithmetic, geometric and {Heinz} means of matrices", journal = j-LIN-MULT-ALGEBRA, volume = "59", number = "12", pages = "1383--1392", year = "2011", CODEN = "LNMLAZ", DOI = "https://doi.org/10.1080/03081087.2011.556632", ISSN = "0308-1087 (print), 1563-5139 (electronic)", ISSN-L = "0308-1087", MRclass = "15A42 (15A45 15B48)", MRnumber = "2855842", MRreviewer = "Yonghui Liu", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Linear and Multilinear Algebra", journal-URL = "http://www.tandfonline.com/loi/glma20", } @Article{Kim:2011:MGM, author = "Sejong Kim and Jimmie Lawson and Yongdo Lim", title = "The matrix geometric mean of parameterized, weighted arithmetic and harmonic means", journal = j-LINEAR-ALGEBRA-APPL, volume = "435", number = "9", pages = "2114--2131", day = "1", month = nov, year = "2011", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2011.04.010", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15B48 (47A64)", MRnumber = "2810556", MRreviewer = "T. Ando", bibdate = "Mon Jun 13 18:34:49 MDT 2011", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; http://www.sciencedirect.com/science/journal/00243795", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @Article{Lecko:2011:DSA, author = "A. Lecko and M. Lecko", title = "Differential subordinations of arithmetic and geometric means of some functionals related to a sector", journal = j-INT-J-MATH-MATH-SCI, pages = "205845:1--205845:19", year = "2011", DOI = "https://doi.org/10.1155/2011/205845", ISSN = "0161-1712 (print), 1687-0425 (electronic)", ISSN-L = "0161-1712", MRclass = "30C80 (30C45)", MRnumber = "2799845", MRreviewer = "Roger W. Barnard", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematics and Mathematical Sciences", journal-URL = "https://www.hindawi.com/journals/ijmms/", } @Article{Long:2011:OGL, author = "Bo-Yong Long and Yu-Ming Chu", title = "Optimal generalized logarithmic mean bounds for the geometric combination of arithmetic and harmonic means", journal = "J. Indones. Math. Soc.", volume = "17", number = "2", pages = "85--96", year = "2011", DOI = "https://doi.org/10.22342/jims.17.2.5.85-95", ISSN = "2086-8952", MRclass = "26E60", MRnumber = "2919042", MRreviewer = "M. Hajja", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of the Indonesian Mathematical Society", } @Article{Molnar:2011:CMM, author = "Lajos Moln{\'a}r", title = "Continuous maps on matrices transforming geometric mean to arithmetic mean", journal = "Ann. Univ. Sci. Budapest. Sect. Comput.", volume = "35", pages = "217--222", year = "2011", ISSN = "0138-9491", MRclass = "47B49 (47A64)", MRnumber = "2894562", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Annales Universitatis Scientiarum Budapestinensis de Rolando E{\"o}tv{\"o}s Nominatae. Sectio Computatorica", } @Article{Shirali:2011:BPG, author = "Shailesh A. Shirali", title = "95.09 {A} bootstrapping proof of the {AM--GM} Inequality for three variables", journal = j-MATH-GAZ, volume = "95", number = "532", pages = "86--87", month = mar, year = "2011", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/S0025557200002412", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Tue May 5 12:04:14 MDT 2015", bibsource = "http://journals.cambridge.org/action/displayIssue?jid=MAG&volumeId=95&issueId=532; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayBackIssues?jid=MAG", } @Article{Wang:2011:SDI, author = "Miao-Kun Wang and Yu-Ming Chu and Gen-Di Wang", title = "A Sharp Double Inequality Between the {Lehmer} and Arithmetic--Geometric Means", journal = j-PAC-J-APPL-MATH, volume = "3", number = "4", pages = "281--286", month = "????", year = "2011", ISSN = "1941-3963", MRclass = "26E60", MRnumber = "3024759", bibdate = "Wed Mar 15 07:14:39 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "https://www.novapublishers.com/catalog/product_info.php?products_id=24770", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Applied Mathematics", journal-URL = "http://www.novapublishers.com/catalog/product_info.php?products_id=6697", } @Article{Zhang:2011:IGM, author = "Qian Zhang and Bing Xu", title = "An invariance of geometric mean with respect to generalized quasi-arithmetic means", journal = j-J-MATH-ANAL-APPL, volume = "379", number = "1", pages = "65--74", year = "2011", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/j.jmaa.2010.12.025", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "26E60 (34A05)", MRnumber = "2776455", MRreviewer = "Alan L. Horwitz", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", } @Article{Albadawi:2012:SVA, author = "Hussien Albadawi", title = "Singular value and arithmetic--geometric mean inequalities for operators", journal = j-ANN-FUNCT-ANAL, volume = "3", number = "1", pages = "10--18", year = "2012", DOI = "https://doi.org/10.15352/afa/1399900020", ISSN = "2008-8752", MRclass = "47A30", MRnumber = "2903264", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Annals of Functional Analysis", journal-URL = "http://projecteuclid.org/afa", } @Article{Aldaz:2012:SBD, author = "J. M. Aldaz", title = "Sharp bounds for the difference between the arithmetic and geometric means", journal = "Arch. Math. (Basel)", volume = "99", number = "4", pages = "393--399", year = "2012", DOI = "https://doi.org/10.1007/s00013-012-0434-7", ISSN = "0003-889x (print), 1420-8938 (electronic)", MRclass = "26E60 (26D15 60E15)", MRnumber = "2990158", MRreviewer = "Huan Nan Shi", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Archiv der Mathematik", } @Article{Barratt:2012:IPC, author = "Carl Barratt and Ramesh Sharma", title = "96.16 {An} inductive proof of the condition for the {AM--GM} equality", journal = j-MATH-GAZ, volume = "96", number = "535", pages = "131--133", month = mar, year = "2012", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/S0025557200004162", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Tue May 5 12:04:21 MDT 2015", bibsource = "http://journals.cambridge.org/action/displayIssue?jid=MAG&volumeId=96&issueId=535; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayBackIssues?jid=MAG", } @Article{Chu:2012:IBA, author = "Yu-Ming Chu and Miao-Kun Wang", title = "Inequalities between arithmetic--geometric, {Gini}, and {Toader} means", journal = j-ABSTR-APPL-ANAL, pages = "830585:1--830585:11", year = "2012", DOI = "https://doi.org/10.1155/2012/830585", ISSN = "1085-3375 (print), 1687-0409 (electronic)", ISSN-L = "1085-3375", MRclass = "26E60 (26D20)", MRnumber = "2861493", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Abstract and Applied Analysis", } @Article{Chu:2012:OLM, author = "Y.-M. Chu and M.-K. Wang and Y.-F. Qiu", title = "Optimal {Lehmer} mean bounds for the geometric and arithmetic combinations of arithmetic and {Seiffert} means", journal = "Azerb. J. Math.", volume = "2", number = "1", pages = "3--9", year = "2012", ISSN = "2218-6816", MRclass = "26E60 (26D99)", MRnumber = "2967278", MRreviewer = "P{\'a}l Burai", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Azerbaijan Journal of Mathematics", } @Article{Chung:2012:VAG, author = "Kun-Jen Chung", title = "On the validity of the arithmetic--geometric mean method to locate the optimal solution in a supply chain system", journal = j-INT-J-SYST-SCI, volume = "43", number = "8", pages = "1454--1463", year = "2012", CODEN = "IJSYA9", DOI = "https://doi.org/10.1080/00207721.2010.547628", ISSN = "0020-7721 (print), 1464-5319 (electronic)", ISSN-L = "0020-7721", MRclass = "90B05", MRnumber = "2946981", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Systems Science. Principles and Applications of Systems and Integration", journal-URL = "http://www.tandfonline.com/loi/tsys20", } @Article{Friedrich:2012:RGM, author = "Jan O. Friedrich and Neill K. J. Adhikari and Joseph Beyene", title = "Ratio of geometric means to analyze continuous outcomes in meta-analysis: comparison to mean differences and ratio of arithmetic means using empiric data and simulation", journal = "Stat. Med.", volume = "31", number = "17", pages = "1857--1886", year = "2012", DOI = "https://doi.org/10.1002/sim.4501", ISSN = "0277-6715", MRclass = "Expansion", MRnumber = "2956005", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Statistics in Medicine", } @Article{Gong:2012:SDI, author = "Wei-Ming Gong and Ying-Qing Song and Miao-Kun Wang and Yu-Ming Chu", title = "A sharp double inequality between {Seiffert}, arithmetic, and geometric means", journal = j-ABSTR-APPL-ANAL, pages = "684834:1--684834:7", year = "2012", ISSN = "1085-3375 (print), 1687-0409 (electronic)", ISSN-L = "1085-3375", MRclass = "26E60 (26D07)", MRnumber = "2965473", MRreviewer = "Raghib M. Abu-Saris", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Abstract and Applied Analysis", } @Article{Hadjidimos:2012:IEO, author = "Apostolos Hadjidimos", title = "Irreducibility and extensions of {Ostrowski's Theorem}", journal = j-LINEAR-ALGEBRA-APPL, volume = "436", number = "7", pages = "2156--2168", year = "2012", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2011.11.035", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379511007786", abstract = "In this paper an extension of Ostrowski's Theorem for complex square irreducible matrices is presented. Also extensions of similar statements for square complex matrices are analyzed and completed. Most of the statements in this work cover also the case of reducible matrices.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Alpha-matrices; Ostrowski's Theorems; H{\"o}lder inequality; (irreducibly) diagonally dominant matrices; Generalized arithmetic--geometric mean inequality", } @Article{He:2012:IAG, author = "Chuanjiang He and Limin Zou and Shahid Qaisar", title = "On improved arithmetic--geometric mean and {Heinz} inequalities for matrices", journal = j-J-MATH-INEQUAL, volume = "6", number = "3", pages = "453--459", year = "2012", DOI = "https://doi.org/10.7153/jmi-06-42", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "15A42 (15A18 15A60)", MRnumber = "3012207", MRreviewer = "Jaspal Singh Aujla", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{He:2012:OIB, author = "Zai-Yin He and Yu-Ming Chu", title = "Optimal inequalities between one-parameter mean and the combination of arithmetic, geometric and harmonic means", journal = j-PAC-J-APPL-MATH, volume = "4", number = "3", pages = "149--154", year = "2012", ISSN = "1941-3963", MRclass = "26E60", MRnumber = "3060210", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Applied Mathematics", journal-URL = "http://www.novapublishers.com/catalog/product_info.php?products_id=6697", } @Article{Kinjo:2012:HSA, author = "Kensaku Kinjo and Yuken Miyasaka", title = "Hypergeometric series and arithmetic--geometric mean over $2$-adic fields", journal = j-INT-J-NUMBER-THEORY, volume = "8", number = "3", pages = "831--844", month = may, year = "2012", DOI = "https://doi.org/10.1142/S1793042112500480", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", MRclass = "11S80 (11G20 33C05)", MRnumber = "2904934", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042112500480", abstract = "Dwork proved that the Gaussian hypergeometric function on $p$-adic numbers can be extended to a function which takes values of the unit roots of ordinary elliptic curves over a finite field of characteristic $ p \geq 3$. We present an analogous theory in the case $ p = 2$. As an application, we give a relation between the canonical lift and the unit root of an elliptic curve over a finite field of characteristic $2$ by using the $2$-adic arithmetic--geometric mean.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @Article{Kittaneh:2012:IAG, author = "Fuad Kittaneh and Mario Krni{\'c} and Neda Lovri{\v{c}}evi{\'c} and Josip Pe{\v{c}}ari{\'c}", title = "Improved arithmetic--geometric and {Heinz} means inequalities for {Hilbert} space operators", journal = j-PUBL-MATH-DEBRECEN, volume = "80", number = "3-4", pages = "465--478", year = "2012", CODEN = "PUMAAR", DOI = "https://doi.org/10.5486/PMD.2012.5193", ISSN = "0033-3883 (print), 2064-2849 (electronic)", MRclass = "47A63 (26D10 26E60)", MRnumber = "2943018", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Publicationes Mathematicae Debrecen", journal-URL = "http://publi.math.unideb.hu/", } @Article{Kum:2012:GMP, author = "Sangho Kum and Yongdo Lim", title = "A geometric mean of parameterized arithmetic and harmonic means of convex functions", journal = j-ABSTR-APPL-ANAL, pages = "836804:1--836804:15", year = "2012", ISSN = "1085-3375 (print), 1687-0409 (electronic)", ISSN-L = "1085-3375", MRclass = "26E60 (26B25)", MRnumber = "3004919", MRreviewer = "Silvia Toader", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Abstract and Applied Analysis", } @Article{Maligranda:2012:GIE, author = "Lech Maligranda", title = "The {AM--GM} Inequality is Equivalent to the {Bernoulli} Inequality", journal = j-MATH-INTEL, volume = "34", number = "1", pages = "1--2", month = "????", year = "2012", CODEN = "MAINDC", DOI = "https://doi.org/10.1007/s00283-011-9266-8", ISSN = "0343-6993 (print), 1866-7414 (electronic)", ISSN-L = "0343-6993", bibdate = "Thu Feb 14 06:39:03 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Mathematical Intelligencer", journal-URL = "http://link.springer.com/journal/283", keywords = "AM (arithmetic mean); GM (geometric mean)", remark = "This paper references two books that provide 52 and 74 proofs of the {\em Cauchy inequality\/} (1821), {$ {\rm AM} \geq {\rm GM} $}, and gives a short proof that it and the {\em Barrow--Bernoulli inequality} (1670, 1689), $ x^n \geq 1 + n(x - 1) $, for any $ x > 0 $ and $n$ an integer, are mutually equivalent: either one implies the other.", } @Article{Maze:2012:NWH, author = "G{\'e}rard Maze and Urs Wagner", title = "A note on the weighted harmonic--geometric--arithmetic means inequalities", journal = j-MATH-INEQUAL-APPL, volume = "15", number = "1", pages = "15--26", year = "2012", DOI = "https://doi.org/10.7153/mia-15-02", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "26D15 (15A42 26E60)", MRnumber = "2919427", MRreviewer = "Alfred Witkowski", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Seo:2012:AGM, author = "Yuki Seo", title = "The arithmetic--geometric mean inequality in an external formula", journal = j-SCI-MATH-JPN, volume = "75", number = "3", pages = "299--305", year = "2012", ISSN = "1346-0862", ISSN-L = "1346-0447", MRclass = "47A63 (26D10)", MRnumber = "3099760", MRreviewer = "T. Ando", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Scientiae Mathematicae Japonicae", journal-URL = "http://www.jams.or.jp/notice/scmj/smj.html", } @Article{Spandaw:2012:HIG, author = "Jeroen Spandaw and Duco van Straten", title = "Hyperelliptic integrals and generalized arithmetic--geometric mean", journal = j-RAMANUJAN-J, volume = "28", number = "1", pages = "61--78", year = "2012", CODEN = "RAJOF9", DOI = "https://doi.org/10.1007/s11139-011-9353-7", ISSN = "1382-4090 (print), 1572-9303 (electronic)", ISSN-L = "1382-4090", MRclass = "14K25 (14H40)", MRnumber = "2914453", MRreviewer = "Ahmed Lesfari", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan", journal-URL = "http://link.springer.com/journal/11139", } @Article{Wang:2012:SDI, author = "Miao-Kun Wang and Yu-Ming Chu and Gen-Di Wang", title = "A Sharp Double Inequality Between the {Lehmer} and Arithmetic--Geometric Means", journal = j-PAC-J-APPL-MATH, volume = "4", number = "1", pages = "1--25", year = "2012", ISSN = "1941-3963", MRclass = "26E60 (26D07)", MRnumber = "3027346", bibdate = "Wed Mar 15 07:16:27 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Applied Mathematics", journal-URL = "http://www.novapublishers.com/catalog/product_info.php?products_id=6697", } @Article{Xia:2012:OOP, author = "Weifeng Xia and Shouwei Hou and Gendi Wang and Yuming Chu", title = "Optimal one-parameter mean bounds for the convex combination of arithmetic and geometric means", journal = "J. Appl. Anal.", volume = "18", number = "2", pages = "197--207", year = "2012", DOI = "https://doi.org/10.1515/jaa-2012-0013", ISSN = "1425-6908", MRclass = "26E60 (26D20)", MRnumber = "2999377", MRreviewer = "Huan Nan Shi", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Applied Analysis", } @Article{Baricz:2013:TTI, author = "{\'A}rp{\'a}d Baricz and Kondooru Raghavendar and Anbhu Swaminathan", title = "{Tur{\'a}n} type inequalities for $q$-hypergeometric functions", journal = j-J-APPROX-THEORY, volume = "168", number = "??", pages = "69--79", year = "2013", CODEN = "JAXTAZ", DOI = "https://doi.org/10.1016/j.jat.2013.01.002", ISSN = "0021-9045 (print), 1096-0430 (electronic)", ISSN-L = "0021-9045", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0021904513000129", abstract = "In this paper our aim is to deduce some Tur{\'a}n type inequalities for $q$-hypergeometric and $q$-confluent hypergeometric functions. In order to obtain the main results we apply the methods developed in the case of classical Kummer and Gauss hypergeometric functions.", acknowledgement = ack-nhfb, fjournal = "Journal of Approximation Theory", journal-URL = "http://www.sciencedirect.com/science/journal/00219045", keywords = "Arithmetic, geometric mean; Basic hypergeometric functions; Gauss and Kummer hypergeometric functions; q -Kummer confluent hypergeometric functions; Tur{\'a}n type inequalities", } @Article{Chu:2013:IAG, author = "Yu Ming Chu and Miao Kun Wang", title = "Inequalities among generalized logarithmic, arithmetic and geometric means", journal = "Acta Math. Sci. Ser. A Chin. Ed.", volume = "33", number = "2", pages = "298--308", year = "2013", ISSN = "1003-3998", MRclass = "26E60 (26D20 41A44)", MRnumber = "3088293", MRreviewer = "Huan Nan Shi", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Acta Mathematica Scientia. Series A. Shuxue Wuli Xuebao. Chinese Edition", } @Article{Chu:2013:STP, author = "Yu-Ming Chu and Miao-Kun Wang and Ye-Fang Qiu and Xiao-Yan Ma", title = "Sharp two parameter bounds for the logarithmic mean and the arithmetic--geometric mean of {Gauss}", journal = j-J-MATH-INEQUAL, volume = "7", number = "3", pages = "349--355", year = "2013", DOI = "https://doi.org/10.7153/jmi-07-31", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26E60", MRnumber = "3115070", MRreviewer = "Biljana P. Mihailovi{\'c}", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Cremona:2013:CAPa, author = "John E. Cremona and Thotsaphon Thongjunthug", title = "The complex {AGM}, periods of elliptic curves over and complex elliptic logarithms", journal = "arXiv.org", volume = "??", number = "??", pages = "1--32", day = "20", month = feb, year = "2013", bibdate = "Tue Mar 14 18:14:33 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "https://arxiv.org/pdf/1011.0914.pdf", abstract = "We give an account of the complex Arithmetic--Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over $C$, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over $C$. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both MAGMA and Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.", acknowledgement = ack-nhfb, } @Article{Cremona:2013:CAPb, author = "John E. Cremona and Thotsaphon Thongjunthug", title = "The complex {AGM}, periods of elliptic curves over {$ \mathbb {C} $} and complex elliptic logarithms", journal = j-J-NUMBER-THEORY, volume = "133", number = "8", pages = "2813--2841", month = aug, year = "2013", CODEN = "JNUTA9", DOI = "https://doi.org/10.1016/j.jnt.2013.02.002", ISSN = "0022-314X (print), 1096-1658 (electronic)", ISSN-L = "0022-314X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022314X13000735", abstract = "We give an account of the complex Arithmetic--Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over $C$, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over $C$. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both MAGMA and Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.", acknowledgement = ack-nhfb, ajournal = "J. Number Theory", fjournal = "Journal of Number Theory", journal-URL = "http://www.sciencedirect.com/science/journal/0022314X", keywords = "Arithmetic--geometric mean; Elliptic curve; Elliptic logarithm; Period lattice", } @Article{Crisan:2013:DSI, author = "O. Cri{\c{s}}an and S. Kanas", title = "Differential subordinations involving arithmetic and geometric means", journal = j-APPL-MATH-COMP, volume = "222", number = "??", pages = "123--131", day = "1", month = oct, year = "2013", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2013.07.051", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", MRclass = "30C45 (30C80)", MRnumber = "3115856", bibdate = "Mon Dec 2 12:34:37 MST 2013", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300313008011", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003/", } @Article{Gumus:2013:IAG, author = "I. Halil Gumus and Necati Taskara", title = "The improved arithmetic--geometric mean inequalities for matrix norms", journal = j-APPL-MATH-SCI-RUSE, volume = "7", number = "29--32", pages = "1439--1446", year = "2013", DOI = "https://doi.org/10.12988/ams.2013.13132", ISSN = "1312-885x (print), 1314-7552 (electronic)", MRclass = "26D15", MRnumber = "3021302", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematical Sciences", journal-URL = "http://www.m-hikari.com/ams/", } @Article{Hassani:2013:AGM, author = "Mehdi Hassani", title = "On the arithmetic--geometric mean inequality", journal = j-TAMKANG-J-MATH, volume = "44", number = "4", pages = "453--456", year = "2013", DOI = "https://doi.org/10.5556/j.tkjm.44.2013.1418", ISSN = "0049-2930 (print), 2073-9826 (electronic)", ISSN-L = "2073-9826", MRclass = "26D15", MRnumber = "3153080", MRreviewer = "M. E. Muldoon", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Tamkang Journal of Mathematics", journal-URL = "http://journals.math.tku.edu.tw/index.php/TKJM", } @Article{Hassani:2013:RAG, author = "Mehdi Hassani", title = "On the ratio of the arithmetic and geometric means of the prime numbers and the number $e$", journal = j-INT-J-NUMBER-THEORY, volume = "9", number = "6", pages = "1593--1603", month = sep, year = "2013", DOI = "https://doi.org/10.1142/S1793042113500450", ISSN = "1793-0421 (print), 1793-7310 (electronic)", ISSN-L = "1793-0421", MRclass = "11N05 (11N56)", MRnumber = "3103906", MRreviewer = "Daniel Fiorilli", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/ijnt.bib", URL = "https://www.worldscientific.com/doi/10.1142/S1793042113500450", abstract = "We study the asymptotic behavior of the sequence with general term consisting of the ratio $ A_n $ by $ G_n $, the arithmetic and geometric means of the prime numbers $ p_1 $, $ p_2 $, \ldots, $ p_n $, respectively, in which, $ p_n $ denotes the $n$-th prime number.", acknowledgement = ack-nhfb, fjournal = "International Journal of Number Theory (IJNT)", journal-URL = "https://www.worldscientific.com/worldscinet/ijnt", } @InCollection{Kinjo:2013:HSA, author = "Kensaku Kinjo and Yuken Miyasaka", booktitle = "Algebraic number theory and related topics 2011", title = "Hypergeometric series and arithmetic--geometric mean over 2-adic fields", publisher = "Res. Inst. Math. Sci. (RIMS), Kyoto", pages = "99--110", year = "2013", MRclass = "11G20 (33C05)", MRnumber = "3221722", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "RIMS K{\^o}ky{\^u}roku Bessatsu, B44", acknowledgement = ack-nhfb, } @Article{Liu:2013:SBS, author = "Baoyu Liu and Weiming Gong and Yingqing Song and Yuming Chu", title = "Sharp bounds for {Seiffert} mean in terms of arithmetic and geometric means", journal = "Int. J. Math. Anal. (Ruse)", volume = "7", number = "33-36", pages = "1765--1773", year = "2013", DOI = "https://doi.org/10.12988/ijma.2013.3349", ISSN = "1312-8876", MRclass = "26E60", MRnumber = "3066546", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Analysis", } @Article{Najafi:2013:SRK, author = "Hamed Najafi", title = "Some results on {Kwong} functions and related inequalities", journal = j-LINEAR-ALGEBRA-APPL, volume = "439", number = "9", pages = "2634--2641", year = "2013", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2013.06.018", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379513004175", abstract = "We investigate some relations between Kwong functions and operator monotone functions. As an application, we present an arithmetic--geometric mean type inequality by showing that for two continuous functions $f$, $g$ on $ (0, \infty)$ such that $ h (t) = f (t) g (t)$ is a Kwong function and $ f (t) g (t) \leq t$, any positive matrices $A$, $B$ and any matrix $X$, it holds that $ ||| f (A) X g (B) + g (A) X f (B) ||| \leq ||| A X + X B |||$ for each unitarily invariant norm $ ||| \cdot |||$.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Arithmetic--geometric mean type inequality; Kwong function; Operator monotone function", } @Article{Ouyang:2013:OPL, author = "Liang-Yuh Ouyang and Chun-Tao Chang", title = "Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging", journal = j-INT-J-PROD-ECON, volume = "144", number = "2", pages = "610--617", year = "2013", CODEN = "JPCEYE", DOI = "https://doi.org/10.1016/j.ijpe.2013.04.027", ISSN = "0925-5273 (print), 1873-7579 (electronic)", ISSN-L = "0925-5273", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0925527313001916", abstract = "The traditional economic production quantity (EPQ) model assumes that the production products are all perfect. It is not always true in the real production system, due to imperfect production process or other factors, imperfect quality items may be produced. Furthermore, it is well-known that the total production-inventory costs can be reduced by reworking the imperfect quality items produced with a relatively smaller additional reworking and holding costs. In addition, the permissible delay in payments offered by the supplier is widely adopted in the practical business market. In this study, we explore the effects of the reworking imperfect quality items and trade credit on the EPQ model with imperfect production processes and complete backlogging. A mathematical model which includes the reworking and shortage costs, interest earned and interest charged is presented. Besides, an arithmetic--geometric mean inequality approach is employed and an algorithm is developed to find the optimal production policy. Furthermore, some numerical examples and sensitivity analysis are provided to demonstrate the proposed model.", acknowledgement = ack-nhfb, fjournal = "International Journal of Production Economics", keywords = "Arithmetic--geometric mean inequality; Complete backlogging; Imperfect production process; Inventory; Permissible delay in payments", } @Article{Shao:2013:SRI, author = "Zhi Hua Shao and Xiao Ming Zhang", title = "Some results involving upper and lower bounds for the difference of arithmetic mean and geometric mean", journal = "Math. Pract. Theory", volume = "43", number = "6", pages = "206--214", year = "2013", ISSN = "1000-0984", MRclass = "26E60", MRnumber = "3114600", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematics in Practice and Theory. Shuxue de Shijian yu Renshi", } @Article{Xu:2013:PPM, author = "Bai-Xiang Xu and Yang Gao and Min-Zhong Wang", title = "Particle packing and the mean theory", journal = j-PHYS-LET-A, volume = "377", number = "3--4", pages = "145--147", year = "2013", CODEN = "PYLAAG", DOI = "https://doi.org/10.1016/j.physleta.2012.11.022", ISSN = "0375-9601 (print), 1873-2429 (electronic)", ISSN-L = "0375-9601", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/kepler.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0375960112011772", abstract = "This Letter presents two mean relations between the densities/porosities of random and regular packing modes. The two mean relations work very well for the packing of spherical particles, cubic particles and circular discs. Results confirm the corresponding experimental and computational results.", acknowledgement = ack-nhfb, fjournal = "Physics Letters A", journal-URL = "http://www.sciencedirect.com/science/journal/03759601", keywords = "Arithmetic--geometric mean; Harmonic--geometric mean; Packing density; Particle packing", } @Article{Yamazaki:2013:EPA, author = "T. Yamazaki", title = "An elementary proof of arithmetic--geometric mean inequality of the weighted {Riemannian} mean of positive definite matrices", journal = j-LINEAR-ALGEBRA-APPL, volume = "438", number = "4", pages = "1564--1569", day = "15", month = feb, year = "2013", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2011.12.006", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15B48", MRnumber = "3005242", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2010.bib", note = "16th \{ILAS\} Conference Proceedings, Pisa 2010", URL = "http://www.sciencedirect.com/science/article/pii/S0024379511007865", abstract = "The weighted Riemannian mean of positive definite matrices is a kind of weighted geometric mean of n matrices. Some properties of the weighted Riemannian mean are easily obtained. But some of them are not easy. In this short paper, a simplified proof of arithmetic--geometric--harmonic mean inequalities will be obtained.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795/", keywords = "Arithmetic--geometric mean inequality; Geometric mean; Matrix inequality; Positive definite matrix; The Riemannian mean", } @Article{Bellissima:2014:AGH, author = "Fabio Bellissima", title = "Arithmetic, geometric and harmonic means in music theory", journal = j-BOLL-STOR-SCI-MAT, volume = "34", number = "2", pages = "201--244", year = "2014", ISSN = "0392-4432 (print), 1724-1650 (electronic)", ISSN-L = "0392-4432", MRclass = "00A65 (01A20 26E60)", MRnumber = "3288578", MRreviewer = "David Warren Bulger", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Bollettino di Storia delle Scienze Matematiche", } @Article{Chang:2014:API, author = "Hung-Chi Chang", title = "An analysis of production-inventory models with deteriorating items in a two-echelon supply chain", journal = j-APPL-MATH-MODEL, volume = "38", number = "3", pages = "1187--1191", year = "2014", CODEN = "AMMODL", DOI = "https://doi.org/10.1016/j.apm.2013.07.031", ISSN = "0307-904x (print), 1872-8480 (electronic)", ISSN-L = "0307-904X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0307904X13004757", abstract = "This paper revisits two previous studies that addressed the integrated production--inventory problem for deteriorating items in a two-echelon supply chain, where the item's deterioration rate is a constant or follows a continuous probability distribution function. The aim of this study is to present an improved solution procedure to determine the delivery lot size and the number of deliveries per production batch cycle that minimizes the total cost of the entire supply chain. The performance of the proposed methodology is illustrated analytically and numerically.", acknowledgement = ack-nhfb, fjournal = "Applied mathematical modelling", keywords = "Arithmetic--geometric mean method; Inventory; Marginal analysis; Probabilistic deterioration; Supply chain", } @Article{Furuichi:2014:OIA, author = "Shigeru Furuichi", title = "Operator inequalities among arithmetic mean, geometric mean and harmonic mean", journal = j-J-MATH-INEQUAL, volume = "8", number = "3", pages = "669--672", year = "2014", DOI = "https://doi.org/10.7153/jmi-08-49", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "47A05 (26E60)", MRnumber = "3260334", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Hassani:2014:AGM, author = "Mehdi Hassani", title = "On the arithmetic--geometric means of positive integers and the number $e$", journal = "Appl. Math. E-Notes", volume = "14", pages = "250--255", year = "2014", ISSN = "1607-2510", MRclass = "26E60", MRnumber = "3324424", MRreviewer = "Bing Xu", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics E-Notes", } @Article{Jameson:2014:AAG, author = "G. J. O. Jameson", title = "An approximation to the arithmetic--geometric mean", journal = j-MATH-GAZ, volume = "98", number = "541", pages = "85--95", month = mar, year = "2014", CODEN = "MAGAAS", DOI = "https://doi.org/10.2307/3621497", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Tue May 5 12:04:31 MDT 2015", bibsource = "http://journals.cambridge.org/action/displayIssue?jid=MAG&volumeId=98&issueId=541; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib; https://www.math.utah.edu/pub/tex/bib/mathgazette2010.bib", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", doi-bad = "https://doi.org/10.2307/3621497", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayBackIssues?jid=MAG", } @Article{Osler:2014:RFR, author = "Thomas J. Osler and Tirupathi R. Chandrupatla", title = "98.23 Recursive formulas related to the {Arithmetic--Geometric Mean}", journal = j-MATH-GAZ, volume = "98", number = "543", pages = "484--486", month = nov, year = "2014", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/S0025557200008202", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Tue Nov 17 09:58:03 MST 2015", bibsource = "http://journals.cambridge.org/action/displayIssue?jid=MAG&volumeId=98&issueId=543; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayBackIssues?jid=MAG", } @Article{Qian:2014:OBN, author = "Wei-Mao Qian and Yu-Ming Chu", title = "Optimal bounds for {Neuman} means in terms of geometric, arithmetic and quadratic means", journal = j-J-INEQUAL-APPL, pages = "175:1--175:13", year = "2014", DOI = "https://doi.org/10.1186/1029-242X-2014-175", ISSN = "1029-242X", MRclass = "26E60", MRnumber = "3346843", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Srivastava:2014:ICE, author = "H. M. Srivastava and N. Magesh and J. Yamini", title = "Initial coefficient estimates for bi-$ \lambda $-convex and bi-$ \mu $-starlike functions connected with arithmetic and geometric means", journal = "Electron. J. Math. Anal. Appl.", volume = "2", number = "2", pages = "152--162", year = "2014", ISSN = "2090-729X", MRclass = "30C45 (30C50)", MRnumber = "3311256", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Electronic Journal of Mathematical Analysis and Applications. EJMAA", } @Article{Villarino:2014:ASP, author = "Mark B. Villarino", title = "The {AGM} Simple Pendulum", journal = "arXiv.org", volume = "??", number = "??", pages = "1--19", day = "1", month = sep, year = "2014", bibdate = "Tue Mar 14 18:11:41 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "https://arxiv.org/pdf/1202.2782.pdf", abstract = "We present a self-contained development of Gauss' Arithmetic--Geometric Mean (AGM) and the work of the great British number theorist A. E. Ingham who obtained rigorous error bounds for the AGM's approximations to the period of a simple pendulum. Moreover we discuss the relation of complex multiplication to the AGM.", acknowledgement = ack-nhfb, remark = "From page 2: ``It is unfortunate that none of the authors [\cite{Carvalhaes:2008:APS} and Alfred George Greenhill \booktitle{The Applications of Elliptic Functions} (1892, 1959)] cites the marvelous investigations of the great British number theorist A. E. [Albert Edward] Ingham [(1900--1967)] which L. A. Pars describes in his monumental 665-page standard work \cite{Pars:1965:TAD,Pars:1968:TAD,Pars:1979:TAD}, which was published almost 50 years ago in 1965. Ingham not only obtains the formulas of Carvalhaes and Suppes [\cite{Carvalhaes:2008:APS}] but also obtains rigorous error estimates, both in excess and in defect. It is beyond question that Ingham's work deserves to be better known.'' [I cannot spot elliptic-function-related work by Ingham in the 28 entries found in the MathSciNet database.]", } @Article{Wang:2014:BPE, author = "Miao-Kun Wang and Yu-Ming Chu and Yue-Ping Jiang and Song-Liang Qiu", title = "Bounds of the perimeter of an ellipse using arithmetic, geometric and harmonic means", journal = j-MATH-INEQUAL-APPL, volume = "17", number = "1", pages = "101--111", year = "2014", DOI = "https://doi.org/10.7153/mia-17-07", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "33E05 (26E60)", MRnumber = "3220978", MRreviewer = "Mehdi Hassani", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Wang:2014:NSC, author = "Hua Wang and Tie-Hong Zhao and Ying-Qing Song and Yu-Ming Chu", title = "Necessary and sufficient conditions for inequalities between the generalized {Muirhead} mean and arithmetic, harmonic and geometric means", journal = j-PAC-J-APPL-MATH, volume = "6", number = "3", pages = "175--187", year = "2014", ISSN = "1941-3963", MRclass = "26E60", MRnumber = "3287241", MRreviewer = "Raghib M. Abu-Saris", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Applied Mathematics", journal-URL = "http://www.novapublishers.com/catalog/product_info.php?products_id=6697", } @Article{Yang:2014:OGC, author = "Lun Yang and Yue-Ying Yang and Qing Wang and Wei-Mao Qian", title = "The optimal geometric combination bounds for {Neuman} means of harmonic, arithmetic and contra-harmonic", journal = j-PAC-J-APPL-MATH, volume = "6", number = "4", pages = "283--292", year = "2014", ISSN = "1941-3963", MRclass = "26E60", MRnumber = "3380234", MRreviewer = "Yu-ming Chu", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Pacific Journal of Applied Mathematics", journal-URL = "http://www.novapublishers.com/catalog/product_info.php?products_id=6697", } @Article{Yang:2014:SBA, author = "Zhen-Hang Yang and Ying-Qing Song and Yu-Ming Chu", title = "Sharp bounds for the arithmetic--geometric mean", journal = j-J-INEQUAL-APPL, pages = "192:1--192:13", year = "2014", DOI = "https://doi.org/10.1186/1029-242X-2014-192", ISSN = "1029-242X", MRclass = "26E60 (26D07 33E05)", MRnumber = "3255894", MRreviewer = "Huan Nan Shi", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", pagecount = "13", } @Article{Yang:2014:SBS, author = "Zhen-Hang Yang", title = "Sharp bounds for {Seiffert} mean in terms of weighted power means of arithmetic mean and geometric mean", journal = j-MATH-INEQUAL-APPL, volume = "17", number = "2", pages = "499--511", year = "2014", DOI = "https://doi.org/10.7153/mia-17-37", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "26E60", MRnumber = "3235026", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Zhao:2014:OII, author = "Jianguo Zhao and Junliang Wu and Haisong Cao and Wenshi Liao", title = "Operator inequalities involving the arithmetic, geometric, {Heinz} and {Heron} means", journal = j-J-MATH-INEQUAL, volume = "8", number = "4", pages = "747--756", year = "2014", DOI = "https://doi.org/10.7153/jmi-08-56", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "47A05 (26D10 26E60)", MRnumber = "3277368", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Zuo:2014:UBS, author = "Hongliang Zuo and Masatoshi Fujii and Jun Ichi Fujii and Yuki Seo", title = "Upper bound for spectra of {Jensen} operator and its application to reverse arithmetic--geometric means", journal = j-MATH-INEQUAL-APPL, volume = "17", number = "2", pages = "641--648", year = "2014", DOI = "https://doi.org/10.7153/mia-17-47", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "47A10 (47A30)", MRnumber = "3235036", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Audenaert:2015:IBA, author = "Koenraad M. R. Audenaert", title = "Interpolating between the arithmetic--geometric mean and {Cauchy--Schwarz} matrix norm inequalities", journal = j-OPER-MATRICES, volume = "9", number = "2", pages = "475--479", year = "2015", DOI = "https://doi.org/10.7153/oam-09-29", ISSN = "1846-3886 (print), 1848-9974 (electronic)", MRclass = "15A60", MRnumber = "3338577", MRreviewer = "Rajendra Bhatia", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Operators and Matrices", journal-URL = "http://oam.ele-math.com/", } @Article{Buric:2015:AEA, author = "Tomislav Buri{\'c} and Neven Elezovi{\'c}", title = "Asymptotic expansion of the arithmetic--geometric mean and related inequalities", journal = j-J-MATH-INEQUAL, volume = "9", number = "4", pages = "1181--1190", year = "2015", DOI = "https://doi.org/10.7153/jmi-09-90", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26E60 (41A60)", MRnumber = "3360162", MRreviewer = "Silvia Toader", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Chu:2015:OBF, author = "Yu-Ming Chu and Wei-Mao Qian and Li-Min Wu and Xiao-Hui Zhang", title = "Optimal bounds for the first and second {Seiffert} means in terms of geometric, arithmetic and contraharmonic means", journal = j-J-INEQUAL-APPL, pages = "44:1--44:9", year = "2015", DOI = "https://doi.org/10.1186/s13660-015-0570-2", ISSN = "1029-242X", MRclass = "26E60", MRnumber = "3305713", MRreviewer = "Raghib M. Abu-Saris", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Gao:2015:DWM, author = "Peng Gao", title = "On a discrete weighted mixed arithmetic--geometric mean inequality", journal = j-MATH-INEQUAL-APPL, volume = "18", number = "3", pages = "941--947", year = "2015", DOI = "https://doi.org/10.7153/mia-18-70", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "26E60 (26D15)", MRnumber = "3344739", MRreviewer = "Eder Kikianty", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Leng:2015:SUB, author = "Tuo Leng and Xiaolin Qin", title = "The sharp upper bound for the ratio between the arithmetic and the geometric mean", journal = j-MATH-INEQUAL-APPL, volume = "18", number = "3", pages = "975--980", year = "2015", DOI = "https://doi.org/10.7153/mia-18-73", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "26E60 (26D15)", MRnumber = "3344742", MRreviewer = "{\'A}d{\'a}m Besenyei", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Nelsen:2015:PWT, author = "Roger B. Nelsen", title = "Proof Without Words: a Trigonometric Proof of the Arithmetic Mean--Geometric Mean Inequality", journal = j-COLLEGE-MATH-J, volume = "46", number = "1", pages = "42--42", month = jan, year = "2015", CODEN = "????", DOI = "https://doi.org/10.4169/college.math.j.46.1.42", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 10:09:33 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.4169/college.math.j.46.1.42", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "27 Nov 2017", } @Article{Nishimura:2015:NPL, author = "Ryo Nishimura", title = "New properties of the lemniscate function and its transformation", journal = j-J-MATH-ANAL-APPL, volume = "427", number = "1", pages = "460--468", year = "2015", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/j.jmaa.2015.02.066", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X15001870", abstract = "In this paper, we show several formulas for the lemniscate function which include an infinite product formula for the lemniscate sine. Furthermore, we show the relation between the product formula and Carlson's algorithm which is known as the variant of the arithmetic geometric mean of Gauss.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "Hypergeometric series; Infinite products; Lemniscate function; Mean iteration", } @Article{Osler:2015:PNR, author = "Thomas J. Osler", title = "A Product of Nested Radicals for the {AGM}", journal = j-AMER-MATH-MONTHLY, volume = "122", number = "9", pages = "886--887", month = nov, year = "2015", CODEN = "AMMYAE", DOI = "https://doi.org/10.4169/amer.math.monthly.122.9.886", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Feb 8 16:33:32 MST 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib", URL = "http://www.jstor.org/stable/10.4169/amer.math.monthly.122.9.886", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Pinelis:2015:EUL, author = "Iosif Pinelis", title = "Exact upper and lower bounds on the difference between the arithmetic and geometric means", journal = j-BULL-AUSTRAL-MATH-SOC, volume = "92", number = "1", pages = "149--158", year = "2015", CODEN = "ALNBAB", DOI = "https://doi.org/10.1017/S0004972715000350", ISSN = "0004-9727 (print), 1755-1633 (electronic)", ISSN-L = "0004-9727", MRclass = "60E15 (26E60)", MRnumber = "3366459", MRreviewer = "Yuanguo Zhu", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Bulletin of the Australian Mathematical Society", journal-URL = "http://journals.cambridge.org/action/displayJournal?jid=BAZ", } @Article{Qian:2015:SBS, author = "Wei-Mao Qian and Yu-Ming Chu and Xiao-Hui Zhang", title = "Sharp bounds for {S{\'a}ndor} mean in terms of arithmetic, geometric and harmonic means", journal = j-J-INEQUAL-APPL, pages = "221:1--221:13", year = "2015", DOI = "https://doi.org/10.1186/s13660-015-0741-1", ISSN = "1029-242X", MRclass = "26E60", MRnumber = "3367707", MRreviewer = "Raghib M. Abu-Saris", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Ruan:2015:NAG, author = "Jiechang Ruan", title = "Notes on the arithmetic--geometric mean inequality for singular values", journal = j-ITAL-J-PURE-APPL-MATH, volume = "35", pages = "227--232", year = "2015", ISSN = "1126-8042 (print), 2239-0227 (electronic)", MRclass = "15A42 (47A63 47B15)", MRnumber = "3477563", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Italian Journal of Pure and Applied Mathematics", journal-URL = "http://ijpam.uniud.it/journal/", } @Article{Xu:2015:RSP, author = "Qian Xu", title = "Research on {Schur}-$p$ power-convexity of the quotient of arithmetic mean and geometric mean", journal = "J. Fudan Univ. Nat. Sci.", volume = "54", number = "3", pages = "288--295", year = "2015", ISSN = "0427-7104", MRclass = "26E60 (26B25)", MRnumber = "3410771", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Fudan University. Journal. Natural Science. Fudan Xuebao. Ziran Kexue Ban", } @Article{Zou:2015:IAG, author = "Limin Zou and Youyi Jiang", title = "Improved arithmetic--geometric mean inequality and its application", journal = j-J-MATH-INEQUAL, volume = "9", number = "1", pages = "107--111", year = "2015", DOI = "https://doi.org/10.7153/jmi-09-10", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "47A63 (26D07 26E60)", MRnumber = "3333909", MRreviewer = "Ali Morassaei", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Zou:2015:RAG, author = "Limin Zou and Yi Huang", title = "A refinement of the arithmetic--geometric mean inequality", journal = j-INT-J-MATH-EDU-SCI-TECH, volume = "46", number = "1", pages = "158--160", year = "2015", CODEN = "IJMEBM", DOI = "https://doi.org/10.1080/0020739X.2014.941426", ISSN = "0020-739x (print), 1464-5211 (electronic)", ISSN-L = "0020-739X", MRclass = "26D15", MRnumber = "3286728", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Education in Science and Technology", journal-URL = "http://www.tandfonline.com/loi/tmes20", } @Article{Zuo:2015:IRA, author = "Hongliang Zuo and Nan Cheng", title = "Improved reverse arithmetic--geometric means inequalities for positive operators on {Hilbert} space", journal = j-MATH-INEQUAL-APPL, volume = "18", number = "1", pages = "51--60", year = "2015", DOI = "https://doi.org/10.7153/mia-18-03", ISSN = "1331-4343 (print), 1848-9966 (electronic)", MRclass = "47A30 (47A63)", MRnumber = "3277056", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Mathematical Inequalities \& Applications", journal-URL = "http://mia.ele-math.com/", } @Article{Adiyasuren:2016:RAG, author = "Vandanjav Adiyasuren and Tserendorj Batbold and Muhammad Adil Khan", title = "Refined arithmetic--geometric mean inequality and new entropy upper bound", journal = j-COMMUN-KOREAN-MATH-SOC, volume = "31", number = "1", pages = "95--100", year = "2016", DOI = "https://doi.org/10.4134/CKMS.2016.31.1.095", ISSN = "1225-1763 (print), 2234-3024 (electronic)", MRclass = "94A17 (26E60)", MRnumber = "3458733", MRreviewer = "Flavia-Corina Mitroi-Symeonidis", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Korean Mathematical Society. Communications", journal-URL = "http://ckms.kms.or.kr/", } @InCollection{Agarwal:2016:BGC, author = "Ravi Agarwal and Hans Agarwal and Syamal Sen", title = "Birth, growth and computation of pi to ten trillion digits (2013)", crossref = "Bailey:2016:PNG", pages = "363--423", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_22", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Almkvist:2016:GLR, author = "Gert Almkvist and Bruce Berndt", title = "{Gauss}, {Landen}, {Ramanujan}, the arithmetic--geometric mean, ellipses, $ \pi $, and the {{\booktitle{Ladies Diary}}} (1988)", crossref = "Bailey:2016:PNG", pages = "125--150", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_8", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Bailey:2016:CDD, author = "David H. Bailey", title = "The computation of $ \pi $ to 29,360,000 decimal digits using {Borweins}' quartically convergent algorithm (1988)", crossref = "Bailey:2016:PNG", pages = "109--124", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_7", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Bailey:2016:CPI, author = "David H. Bailey and Jonathan M. Borwein and Andrew Mattingly and Glenn Wightwick", title = "The computation of previously inaccessible digits of $ \pi $", crossref = "Bailey:2016:PNG", pages = "327--339", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_20", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Bailey:2016:RCV, author = "David H. Bailey and Peter B. Borwein and Simon Plouffe", title = "On the rapid computation of various polylogarithmic constants (1997)", crossref = "Bailey:2016:PNG", pages = "219--231", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_?", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Bakherad:2016:RRG, author = "Mojtaba Bakherad", title = "Refinements of a reversed {AM--GM} operator inequality", journal = j-LIN-MULT-ALGEBRA, volume = "64", number = "9", pages = "1687--1695", year = "2016", CODEN = "LNMLAZ", DOI = "https://doi.org/10.1080/03081087.2015.1114984", ISSN = "0308-1087 (print), 1563-5139 (electronic)", ISSN-L = "0308-1087", bibdate = "Tue Sep 20 15:15:01 MDT 2016", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linmultalgebra.bib", acknowledgement = ack-nhfb, fjournal = "Linear Multilinear Algebra", journal-URL = "http://www.tandfonline.com/loi/glma20", onlinedate = "01 Dec 2015", } @InCollection{Borwein:2016:AGM, author = "J. M. Borwein and P. B. Borwein", title = "The arithmetic--geometric mean and fast computation of elementary functions (1984)", crossref = "Bailey:2016:PNG", pages = "79--96", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_4", bibdate = "Thu Aug 11 09:36:22 MDT 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://link.springer.com/chapter/10.1007/978-3-319-32377-0_4", acknowledgement = ack-nhfb, } @InCollection{Borwein:2016:RME, author = "Jonathan M. Borwein and Peter B. Borwein and David H. Bailey", title = "{Ramanujan}, modular equations, and approximations to pi or how to compute one billion digits of pi (1989)", crossref = "Bailey:2016:PNG", pages = "175--195", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_?", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Brent:2016:FMP, author = "Richard P. Brent", title = "Fast multiple-precision evaluation of elementary functions (1976)", crossref = "Bailey:2016:PNG", pages = "9--20", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_2", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Buric:2016:AAI, author = "Tomislav Buri{\'c}", title = "Asymptotic analysis of the iterative power means", journal = j-J-MATH-ANAL-APPL, volume = "433", number = "1", pages = "701--705", year = "2016", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/j.jmaa.2015.08.020", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X15007477", abstract = "We investigate the asymptotic expansion of the compound mean obtained by the iterative process of two power means. We present the stationary and convergence properties of the coefficients in the expansions.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "Arithmetic--geometric mean; Asymptotic expansion; Compound mean; Power mean", } @Article{Cardoso:2016:MAG, author = "Jo{\~a}o R. Cardoso and Rui Ralha", title = "Matrix Arithmetic--Geometric Mean and the Computation of the Logarithm", journal = j-SIAM-J-MAT-ANA-APPL, volume = "37", number = "2", pages = "719--743", month = "????", year = "2016", CODEN = "SJMAEL", DOI = "https://doi.org/10.1137/140998226", ISSN = "0895-4798 (print), 1095-7162 (electronic)", ISSN-L = "0895-4798", MRclass = "65F60 (33E05 65F30)", MRnumber = "3507553", MRreviewer = "Volker Karl Richard Grimm", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/2; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Matrix Analysis and Applications", journal-URL = "http://epubs.siam.org/simax", onlinedate = "January 2016", } @InCollection{Cox:2016:AGM, author = "David A. Cox", title = "The arithmetic--geometric mean of {Gauss} (1984)", crossref = "Bailey:2016:PNG", pages = "21--78", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_3", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Fujii:2016:RDM, author = "Jun Ichi Fujii and Masatoshi Fujii and Yuki Seo and Hongliang Zuo", title = "Recent developments of matrix versions of the arithmetic--geometric mean inequality", journal = j-ANN-FUNCT-ANAL, volume = "7", number = "1", pages = "102--117", year = "2016", DOI = "https://doi.org/10.1215/20088752-3429400", ISSN = "2008-8752", MRclass = "47A63 (15A18 15A42 47-02 47A30)", MRnumber = "3449343", MRreviewer = "Rajendra Bhatia", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Annals of Functional Analysis", journal-URL = "http://projecteuclid.org/afa", } @Article{Guo:2016:SBN, author = "Zhi-Jun Guo and Yan Zhang and Yu-Ming Chu and Ying-Qing Song", title = "Sharp bounds for {Neuman} means in terms of geometric, arithmetic and quadratic means", journal = j-J-MATH-INEQUAL, volume = "10", number = "2", pages = "301--312", year = "2016", DOI = "https://doi.org/10.7153/jmi-10-25", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26E60", MRnumber = "3455365", MRreviewer = "Wei-Dong Jiang", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Hirzallah:2016:SVC, author = "Omar Hirzallah", title = "Singular values of convex functions of operators and the arithmetic--geometric mean inequality", journal = j-J-MATH-ANAL-APPL, volume = "433", number = "2", pages = "935--947", year = "2016", CODEN = "JMANAK", DOI = "https://doi.org/10.1016/j.jmaa.2015.08.036", ISSN = "0022-247x (print), 1096-0813 (electronic)", ISSN-L = "0022-247X", MRclass = "47A63", MRnumber = "3398744", MRreviewer = "Eizaburo Kamei", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0022247X15007635", abstract = "We prove singular value inequalities for convex functions of products and sums of operators that generalize the arithmetic--geometric mean inequality for operators. Among other results, we prove that if $ A_i, B_i, X_i, Y_i $, $ i = 1, \ldots, n $ are operators on a complex separable Hilbert space such that $ | X_i | 2 + | Y_i | 2 2 n \leq I $, $ i = 1, \ldots, n $ and if $f$ is a nonnegative increasing convex function on $ [0, \infty)$ satisfying $ f (0) = 0$, then $ s_j (f (| \sum_{i = 1}^n A_i X_i Y_i^\star B_i^\star |)) \leq (1 / 2) s_j (\oplus_{k = 1}^n (X_k^\star (\sum_{i = 1}^n f(| A_i^\star A_k |)) X_k + Y_k^\star (\sum_{i = 1}^n f(| B_i^\star B_k |)) Y_k))$ for $ j = 1, 2, \ldots $.", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Analysis and Applications", journal-URL = "http://www.sciencedirect.com/science/journal/0022247X", keywords = "Compact operator; Convex function; Inequality; Positive operator; Singular value", } @Article{Hoffmann:2016:WGI, author = "Heiko Hoffmann", title = "Weighted {AM--GM} Inequality via Elementary Multivariable Calculus", journal = j-COLLEGE-MATH-J, volume = "47", number = "1", pages = "56--58", month = jan, year = "2016", CODEN = "????", DOI = "https://doi.org/10.4169/college.math.j.47.1.56", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Thu Feb 14 10:09:43 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/abs/10.4169/college.math.j.47.1.56", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "27 Nov 2017", } @Article{Israel:2016:AGM, author = "Arie Israel and Felix Krahmer and Rachel Ward", title = "An arithmetic--geometric mean inequality for products of three matrices", journal = j-LINEAR-ALGEBRA-APPL, volume = "488", number = "??", pages = "1--12", day = "1", month = jan, year = "2016", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2015.09.013", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A45", MRnumber = "3419770", MRreviewer = "Tanvi Jain", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379515005285", abstract = "Consider the following noncommutative arithmetic--geometric mean inequality: Given positive-semidefinite matrices $ A_1, \ldots, A_n $, the following holds for each integer $ m \leq n $: $ (1 / n^m) \sum_{j_1, j_2, \ldots, j_m = 1}^n ||| A_{j_1} A_{j_2} \ldots A_{j_m} ||| \geq ((n - m)! / n!) \sum_{j_1, j_2, \ldots, j_m} = 1 ({\rm all distinct})^n ||| A_{j_1} A_{j_2} \ldots A_{j_m} ||| $, where $ ||| \cdot ||| $ denotes a unitarily invariant norm, including the operator norm and Schatten $p$-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, $ m \leq 3$. The proofs for $ m = 1, 2$ are straightforward; to derive the proof for $ m = 3$, we appeal to a variant of the classic Araki--Lieb--Thirring inequality for permutations of matrix products.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795/", keywords = "Arithmetic--geometric mean inequality; Linear algebra; Norm inequalities", } @InCollection{Kanada:2016:VMP, author = "Yasumasa Kanada", title = "Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation (1988)", crossref = "Bailey:2016:PNG", pages = "151--164", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_9", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Newman:2016:SVF, author = "D. J. Newman", title = "A simplified version of the fast algorithms of {Brent} and {Salamin} (1985)", crossref = "Bailey:2016:PNG", pages = "97--102", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_5", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @InCollection{Salamin:2016:CUA, author = "Eugene Salamin", title = "Computation of $ \pi $ using arithmetic--geometric mean (1976)", crossref = "Bailey:2016:PNG", pages = "1--8", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0_1", bibdate = "Tue Mar 14 11:34:57 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, } @Article{Sheikhhosseini:2016:NRV, author = "Alemeh Sheikhhosseini", title = "A numerical radius version of the arithmetic--geometric mean of operators", journal = "Filomat", volume = "30", number = "8", pages = "2139--2145", year = "2016", DOI = "https://doi.org/10.2298/FIL1608139S", ISSN = "0354-5180", MRclass = "47A30 (47A12)", MRnumber = "3583150", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Univerzitet u Ni{\v{s}}u. Prirodno-Matemati{\v{c}}ki Fakultet. Filomat", } @Misc{Singh:2016:PAL, author = "Paramanand Singh", title = "$ \pi $ ({PI}) and the {AGM}: {Legendre}'s Identity", howpublished = "Web site", year = "2016", bibdate = "Tue Mar 14 10:19:25 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "https://paramanands.blogspot.com/2009/08/pi-and-the-agm-legendres-identity.html", acknowledgement = ack-nhfb, } @Article{Wang:2016:OBG, author = "Hua Wang and Wei-Mao Qian and Yu-Ming Chu", title = "Optimal bounds for {Gaussian} arithmetic--geometric mean with applications to complete elliptic integral", journal = j-FUNCT-SPACES, pages = "3698463:1--3698463:6", year = "2016", DOI = "https://doi.org/10.1155/2016/3698463", ISSN = "2314-8896", MRclass = "26E60 (33E05)", MRnumber = "3531324", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Function Spaces", journal-URL = "https://www.hindawi.com/journals/jfs/", } @Article{Zou:2016:NIB, author = "Limin Zou and Youyi Jiang", title = "A note on interpolation between the arithmetic--geometric mean and {Cauchy--Schwarz} matrix norm inequalities", journal = j-J-MATH-INEQUAL, volume = "10", number = "4", pages = "1119--1122", year = "2016", DOI = "https://doi.org/10.7153/jmi-10-88", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "15A60 (47A63)", MRnumber = "3581159", MRreviewer = "Mitsuru Uchiyama", bibdate = "Tue Mar 14 07:52:56 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @TechReport{Brent:2017:JBP, author = "Richard P. Brent", title = "{Jonathan Borwein}, Pi and the {AGM}", type = "Talk slides", institution = "Australian National University and CARMA, University of Newcastle", address = "Canberra, ACT and Newcastle, NSW, Australia", pages = "76", day = "26", month = sep, year = "2017", bibdate = "Fri Sep 04 17:08:54 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", URL = "https://carma.newcastle.edu.au/meetings/jbcc/abstracts/pdf/JBCC-Richard_Brent.pdf", abstract = "We consider some of Jon Borwein s contributions to the high-precision computation of $ \pi $ and the elementary functions, with particular reference to the fascinating book \booktitle{Pi and the AGM}(Wiley, 1987) by Jon and his brother Peter Borwein. Here ``AGM'' is the arithmetic-geometric mean, first studied by Euler, Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $ \pi $, and more generally the elementary functions. These algorithms run in ``almost linear' time $ O(M(n) \log n)$, where $ M(n)$ is the time for $n$-bit multiplication. The talk will survey some of the results and algorithms, from the time of Archimedes to the present day, that were of interest to Jon. In several cases they were discovered or improved by him", acknowledgement = ack-nhfb, ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", } @Article{Buric:2017:CAA, author = "Tomislav Buri{\'c} and Neven Elezovi{\'c}", title = "Computation and analysis of the asymptotic expansions of the compound means", journal = j-APPL-MATH-COMP, volume = "303", number = "??", pages = "48--54", year = "2017", CODEN = "AMHCBQ", DOI = "https://doi.org/10.1016/j.amc.2017.01.025", ISSN = "0096-3003 (print), 1873-5649 (electronic)", ISSN-L = "0096-3003", bibdate = "Tue Mar 14 16:13:28 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0096300317300334", abstract = "We derive algorithms for computing asymptotic expansion of the composite mean of two arbitrary means M and N. Then we analyse asymptotic behaviour of the compound mean $ M \otimes N $. Examples and application to some classical means are also presented.", acknowledgement = ack-nhfb, fjournal = "Applied Mathematics and Computation", journal-URL = "http://www.sciencedirect.com/science/journal/00963003", keywords = "Arithmetic--geometric mean; Asymptotic expansion; Compound mean; Power mean", } @Article{Chang:2017:AGM, author = "Chun-Tao Chang and Liang-Yuh Ouyang", title = "An arithmetic--geometric mean inequality approach for determining the optimal production lot size with backlogging and imperfect rework process", journal = "J. Appl. Anal. Comput.", volume = "7", number = "1", pages = "224--235", year = "2017", ISSN = "2156-907X", MRclass = "90B05", MRnumber = "3528209", MRreviewer = "Jian-Teng Xu", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Journal of Applied Analysis and Computation", } @Article{Choi:2017:IRA, author = "D. Choi and M. Sababheh", title = "Inequalities related to the arithmetic, geometric and harmonic means", journal = j-J-MATH-INEQUAL, volume = "11", number = "1", pages = "1--16", year = "2017", DOI = "https://doi.org/10.7153/jmi-11-01", ISSN = "1846-579x (print), 1848-9575 (electronic)", MRclass = "26E60 (47A63)", MRnumber = "3601921", MRreviewer = "Wei-Dong Jiang", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Mathematical Inequalities", journal-URL = "http://jmi.ele-math.com/", } @Article{Ding:2017:OBA, author = "Qing Ding and Tiehong Zhao", title = "Optimal bounds for arithmetic--geometric and {Toader} means in terms of generalized logarithmic mean", journal = j-J-INEQUAL-APPL, pages = "102:1--102:12", year = "2017", DOI = "https://doi.org/10.1186/s13660-017-1365-4", ISSN = "1029-242X", MRclass = "26E60", MRnumber = "3647198", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Journal of Inequalities and Applications", journal-URL = "http://journalofinequalitiesandapplications.springeropen.com/", } @Article{Griffiths:2017:AGM, author = "Martin Griffiths and Des MacHale", title = "On arithmetic--geometric-mean polynomials", journal = j-INT-J-MATH-EDU-SCI-TECH, volume = "48", number = "1", pages = "111--117", year = "2017", CODEN = "IJMEBM", DOI = "https://doi.org/10.1080/0020739X.2016.1172740", ISSN = "0020-739x (print), 1464-5211 (electronic)", ISSN-L = "0020-739X", MRclass = "26D05 (26E60)", MRnumber = "3580860", MRreviewer = "Zhen-Hang Yang", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "International Journal of Mathematical Education in Science and Technology", journal-URL = "http://www.tandfonline.com/loi/tmes20", } @Article{Iannazzo:2017:RBB, author = "Bruno Iannazzo and Margherita Porcelli", title = "The {Riemannian Barzilai--Borwein} method with nonmonotone line search and the matrix geometric mean computation", journal = j-IMA-J-NUMER-ANAL, volume = "38", number = "1", pages = "495--517", month = apr, year = "2017", CODEN = "IJNADH", DOI = "https://doi.org/10.1093/imanum/drx015", ISSN = "0272-4979 (print), 1464-3642 (electronic)", ISSN-L = "0272-4979", bibdate = "Fri Feb 9 09:25:58 2018", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "IMA Journal of Numerical Analysis", journal-URL = "http://imajna.oxfordjournals.org/content/by/year", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", } @Article{Rodin:2017:VIA, author = "Burt Rodin", title = "Variance and the inequality of arithmetic and geometric means", journal = j-ROCKY-MOUNTAIN-J-MATH, volume = "47", number = "2", pages = "637--648", year = "2017", CODEN = "RMJMAE", DOI = "https://doi.org/10.1216/RMJ-2017-47-2-637", ISSN = "0035-7596 (print), 1945-3795 (electronic)", ISSN-L = "0035-7596", MRclass = "26D07 (26E60)", MRnumber = "3635378", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "The Rocky Mountain Journal of Mathematics", journal-URL = "http://projecteuclid.org/euclid.rmjm", } @Article{Sawhney:2017:TPG, author = "Mehtaab S. Sawhney", title = "A Telescoping Proof of the {AM--GM} Inequality", journal = j-AMER-MATH-MONTHLY, volume = "124", number = "4", pages = "356--356", month = apr, year = "2017", CODEN = "AMMYAE", DOI = "https://doi.org/10.4169/amer.math.monthly.124.4.356", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Oct 30 07:18:01 MDT 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib", URL = "http://www.jstor.org/stable/10.4169/amer.math.monthly.124.4.356", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html", } @Article{Sheikhhosseini:2017:AGM, author = "Alemeh Sheikhhosseini", title = "An arithmetic--geometric mean inequality related to numerical radius of matrices", journal = "Konuralp J. Math.", volume = "5", number = "1", pages = "85--91", year = "2017", ISSN = "2147-625X", MRclass = "15A60 (26E60)", MRnumber = "3637699", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Konuralp Journal of Mathematics", } @Article{Wu:2017:NYA, author = "Yanqiu Wu", title = "Note on {Young} and arithmetic--geometric mean inequalities for matrices", journal = j-ITAL-J-PURE-APPL-MATH, volume = "37", pages = "347--350", year = "2017", ISSN = "1126-8042 (print), 2239-0227 (electronic)", MRclass = "15A42", MRnumber = "3622936", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Italian Journal of Pure and Applied Mathematics", journal-URL = "http://ijpam.uniud.it/journal/", } @Article{Xue:2017:RWA, author = "Jianming Xue", title = "On reverse weighted arithmetic--geometric mean inequalities for two positive operators", journal = j-ITAL-J-PURE-APPL-MATH, volume = "37", pages = "113--116", year = "2017", ISSN = "1126-8042 (print), 2239-0227 (electronic)", MRclass = "47A63 (47A30)", MRnumber = "3622918", bibdate = "Tue Aug 15 09:24:34 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", acknowledgement = ack-nhfb, fjournal = "Italian Journal of Pure and Applied Mathematics", journal-URL = "http://ijpam.uniud.it/journal/", } @Article{Zou:2017:AGM, author = "Limin Zou", title = "An arithmetic--geometric mean inequality for singular values and its applications", journal = j-LINEAR-ALGEBRA-APPL, volume = "528", number = "??", pages = "25--32", day = "1", month = sep, year = "2017", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2016.01.016", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", MRclass = "15A42 (47A63)", MRnumber = "3652834", bibdate = "Tue Aug 15 10:10:06 2017", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379516000185", abstract = "In this short note, we give a new equivalent form of the arithmetic--geometric mean inequality for singular values. As applications of our result, we give a new proof of an inequality due to Bhatia and Davis (1993) [4] and we obtain a singular value inequality for matrix means, which is similar to one proved by Drury (2012) [9]. Finally, we present a log-majorization inequality for singular values.", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", keywords = "Matrix means; Singular values; Unitarily invariant norms", } @Article{Bertot:2018:DDP, author = "Yves Bertot and Laurence Rideau and Laurent Th{\'e}ry", title = "Distant Decimals of $ \pi $: Formal Proofs of Some Algorithms Computing Them and Guarantees of Exact Computation", journal = j-J-AUTOM-REASON, volume = "61", number = "1--4", pages = "33--71", month = jun, year = "2018", CODEN = "JAREEW", DOI = "https://doi.org/10.1007/s10817-017-9444-2", ISSN = "0168-7433 (print), 1573-0670 (electronic)", ISSN-L = "0168-7433", bibdate = "Sat Aug 4 07:51:41 MDT 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", URL = "http://link.springer.com/article/10.1007/s10817-017-9444-2", acknowledgement = ack-nhfb, fjournal = "Journal of Automated Reasoning", journal-URL = "http://link.springer.com/journal/10817", keywords = "Arithmetic geometric means; Bailey, Borwein, and Plouffe formula; BBP; Coq proof assistant; Formal proofs in real analysis; PI", } @Article{Gencev:2018:PIB, author = "Marian Gencev", title = "On a Proof of the Inequality Between the Arithmetic and Geometric Means", journal = j-AMER-MATH-MONTHLY, volume = "125", number = "7", pages = "650--652", year = "2018", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2018.1470422", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Dec 13 17:59:05 MST 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2010.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "03 Aug 2018", } @Article{Jameson:2018:RNA, author = "G. J. O. Jameson", title = "102.45 {Revisiting} a note on arithmetic and geometric means", journal = j-MATH-GAZ, volume = "102", number = "555", pages = "513", month = nov, year = "2018", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/mag.2018.125", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Thu Feb 14 07:32:40 MST 2019", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib", URL = "https://www.cambridge.org/core/journals/mathematical-gazette/article/10245-revisiting-a-note-on-arithmetic-and-geometric-means/7D7E5F6E956573FB16FCB8998A9E3193", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620", onlinedate = "17 October 2018", } @Article{Zou:2019:UAG, author = "Limin Zou", title = "Unification of the arithmetic-geometric mean and {H{\"o}lder} inequalities for unitarily invariant norms", journal = j-LINEAR-ALGEBRA-APPL, volume = "562", number = "??", pages = "154--162", day = "1", month = feb, year = "2019", CODEN = "LAAPAW", DOI = "https://doi.org/10.1016/j.laa.2018.09.030", ISSN = "0024-3795 (print), 1873-1856 (electronic)", ISSN-L = "0024-3795", bibdate = "Thu Dec 27 15:29:01 MST 2018", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/linala2015.bib", URL = "http://www.sciencedirect.com/science/article/pii/S0024379518304713", acknowledgement = ack-nhfb, fjournal = "Linear Algebra and its Applications", journal-URL = "http://www.sciencedirect.com/science/journal/00243795", } @InProceedings{Brent:2020:BBP, author = "Richard P. Brent", title = "The {Borwein} Brothers, Pi and the {AGM}", crossref = "Bailey:2020:AVC", pages = "323--347", year = "2020", DOI = "https://doi.org/10.1007/978-3-030-36568-4_21", bibdate = "Tue Apr 21 10:54:18 MDT 2020", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", } @Article{Graham:2020:EPA, author = "Cole Graham and Tadashi Tokieda", title = "An Entropy Proof of the Arithmetic Mean-Geometric Mean Inequality", journal = j-AMER-MATH-MONTHLY, volume = "127", number = "6", pages = "545--546", year = "2020", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2020.1738827", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Dec 13 15:45:46 MST 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "21 May 2020", } @Article{Hajja:2020:MPG, author = "Mowaffaq Hajja", title = "104.17 {More} proofs of the {AM--GM} inequality", journal = j-MATH-GAZ, volume = "104", number = "560", pages = "318--326", month = jul, year = "2020", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/mag.2020.59", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Mon Aug 10 09:47:34 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2020.bib", URL = "https://www.cambridge.org/core/journals/mathematical-gazette/article/10417-more-proofs-of-the-amgm-inequality/089FD8AFA839D001F8C8BDC97E438896", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620", onlinedate = "18 June 2020", } @Article{Mahmoudi:2020:GIG, author = "M. G. Mahmoudi", title = "The {AM--GM} Inequality via Gradient", journal = j-COLLEGE-MATH-J, volume = "51", number = "2", pages = "141--143", year = "2020", CODEN = "????", DOI = "https://doi.org/10.1080/07468342.2020.1697605", ISSN = "0746-8342 (print), 1931-1346 (electronic)", ISSN-L = "0746-8342", bibdate = "Mon Mar 9 11:58:57 MDT 2020", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/collegemathj.bib", URL = "http://www.tandfonline.com/doi/full/10.1080/07468342.2020.1697605", acknowledgement = ack-nhfb, fjournal = "College Mathematics Journal", journal-URL = "https://maa.tandfonline.com/loi/ucmj20; https://www.jstor.org/journal/collmathj", onlinedate = "25 Feb 2020", } @Article{Plaza:2020:HLA, author = "Angel Plaza", title = "Harmonic, Logarithmic, and Arithmetic Means and Corollaries", journal = j-AMER-MATH-MONTHLY, volume = "127", number = "5", pages = "427--427", year = "2020", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2020.1726705", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Mon Dec 13 15:45:45 MST 2021", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "23 Apr 2020", } @Article{Moustrou:2022:SRG, author = "Philippe Moustrou and Helen Naumann and Cordian Riener and Thorsten Theobald and Hugues Verdure", title = "Symmetry Reduction in {AM/GM}-Based Optimization", journal = j-SIAM-J-OPT, volume = "32", number = "2", pages = "765--785", month = "????", year = "2022", CODEN = "SJOPE8", DOI = "https://doi.org/10.1137/21M1405691", ISSN = "1052-6234 (print), 1095-7189 (electronic)", ISSN-L = "1052-6234", bibdate = "Tue Oct 17 12:21:37 MDT 2023", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SIOPT/32/2; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/siamjopt.bib", URL = "https://epubs.siam.org/doi/10.1137/21M1405691", acknowledgement = ack-nhfb, fjournal = "SIAM Journal on Optimization", journal-URL = "http://epubs.siam.org/siopt", } @Article{Plaza:2022:FBP, author = "{\'A}ngel Plaza", title = "106.07 {A} function-based proof of the harmonic mean --- geometric mean --- arithmetic mean inequalities", journal = j-MATH-GAZ, volume = "106", number = "565", pages = "130--131", month = mar, year = "2022", CODEN = "MAGAAS", DOI = "https://doi.org/10.1017/mag.2022.22", ISSN = "0025-5572 (print), 2056-6328 (electronic)", ISSN-L = "0025-5572", bibdate = "Mon Jul 18 08:23:47 MDT 2022", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2020.bib", URL = "https://www.cambridge.org/core/journals/mathematical-gazette/article/10607-a-functionbased-proof-of-the-harmonic-mean-geometric-mean-arithmetic-mean-inequalities/61AA72E9B6B3A99B4A12F6C768665CFB", acknowledgement = ack-nhfb, ajournal = "Math. Gaz.", fjournal = "The Mathematical Gazette", journal-URL = "http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620", onlinedate = "24 February 2022", } @Article{Griffin:2023:AJS, author = "Michael J. Griffin and Ken Ono and Neelam Saikia and Wei-Lun Tsai", title = "{AGM} and Jellyfish Swarms of Elliptic Curves", journal = j-AMER-MATH-MONTHLY, volume = "130", number = "4", pages = "355--369", year = "2023", CODEN = "AMMYAE", DOI = "https://doi.org/10.1080/00029890.2022.2160157", ISSN = "0002-9890 (print), 1930-0972 (electronic)", ISSN-L = "0002-9890", bibdate = "Fri Aug 25 08:24:36 MDT 2023", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2020.bib", acknowledgement = ack-nhfb, fjournal = "American Mathematical Monthly", journal-URL = "http://www.jstor.org/journals/00029890.html; https://www.tandfonline.com/loi/uamm20", onlinedate = "16 Feb 2023", }

%%% ==================================================================== %%% Cross-referenced entries must come last:

@Proceedings{Traub:1976:ACC, editor = "J. F. (Joseph Frederick) Traub", booktitle = "{Analytic computational complexity: Proceedings of the Symposium on Analytic Computational Complexity, held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7--8, 1975}", title = "{Analytic computational complexity: Proceedings of the Symposium on Analytic Computational Complexity, held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7--8, 1975}", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "ix + 239", year = "1976", ISBN = "0-12-697560-4", ISBN-13 = "978-0-12-697560-4", LCCN = "QA297 .S915 1975", bibdate = "Sun Dec 30 18:48:22 MST 2007", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", acknowledgement = ack-nhfb, meetingname = "Symposium on Analytic Computational Complexity, Carnegie-Mellon University, 1975.", remark = "", subject = "Numerical analysis; Data processing; Congresses; Computational complexity", } @Proceedings{Martin:1988:SPN, editor = "Joanne L. Martin and Stephen F. Lundstrom", booktitle = "Supercomputing '88: proceedings, November 14--18, 1988, Orlando, Florida", title = "Supercomputing '88: proceedings, November 14--18, 1988, Orlando, Florida", volume = "2", publisher = pub-IEEE, address = pub-IEEE:adr, pages = "viii + 263", year = "1988", ISBN = "0-8186-0882-X (v. 1; paper), 0-8186-8882-3 (v. 1; case), 0-8186-4882-1 (v. 1: microfiche) 0-8186-8923-4 (v. 2), 0-8186-5923-X (v. 2: microfiche), 0-8186-8923-4 (v. 2: case)", ISBN-13 = "978-0-8186-0882-7 (v. 1; paper), 978-0-8186-8882-9 (v. 1; case), 978-0-8186-4882-3 (v. 1: microfiche) 978-0-8186-8923-9 (v. 2), 978-0-8186-5923-2 (v. 2: microfiche), 978-0-8186-8923-9 (v. 2: case)", LCCN = "QA76.5 .S894 1988", bibdate = "Fri Aug 30 08:01:51 MDT 1996", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", note = "Two volumes. IEEE catalog number 88CH2617-9. IEEE Computer Society Order Number 882.", acknowledgement = ack-nhfb, classification = "C5440 (Multiprocessor systems and techniques); C7300 (Natural sciences)", keywords = "biology computing; chemistry; computational biology; computational fluid dynamics; computational mathematics; computational physics; flow simulation; global change; mathematics computing; parallel processing; physics computing; structural analysis; structural engineering computing; supercomputers --- congresses", } @Proceedings{Adams:1993:SCA, editor = "E. Adams and U. Kulisch", booktitle = "Scientific computing with automatic result verification", title = "Scientific computing with automatic result verification", volume = "189", publisher = pub-ACADEMIC, address = pub-ACADEMIC:adr, pages = "x + 612", year = "1993", ISBN = "0-12-044210-8", ISBN-13 = "978-0-12-044210-2", LCCN = "QA76.S368 1993", bibdate = "Mon Dec 18 10:27:38 1995", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib", series = "Mathematics in science and engineering", acknowledgement = ack-nhfb, } @Proceedings{Cohen:1995:AAA, editor = "G. (G{\'e}rard) Cohen and Marc Giusti and Teo Mora", booktitle = "{Applied algebra, algebraic algorithms, and error-correcting codes: 11th international symposium, AAECC-11, Paris, France, July 1995: proceedings}", title = "{Applied algebra, algebraic algorithms, and error-correcting codes: 11th international symposium, AAECC-11, Paris, France, July 1995: proceedings}", volume = "948", publisher = pub-SV, address = pub-SV:adr, pages = "xi + 484", year = "1995", ISBN = "3-540-60114-7 (softcover)", ISBN-13 = "978-3-540-60114-2 (softcover)", LCCN = "QA268 .A35 1995", bibdate = "Tue Mar 14 15:37:05 MDT 2017", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; z3950.loc.gov:7090/Voyager", series = "Lecture notes in computer science", URL = "http://www.loc.gov/catdir/enhancements/fy0815/95021560-d.html", acknowledgement = ack-nhfb, meetingname = "AAECC-11 (1995: Paris, France)", subject = "Error-correcting codes (Information theory); Congresses; Algebra; Data processing; Algorithms", } @Proceedings{Alefeld:1996:SCV, editor = "G{\"o}tz Alefeld and Andreas Frommer and Bruno Lang", booktitle = "Scientific computing and validated numerics: proceedings of the International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics SCAN-95 held in Wuppertal, Germany, September 26--29, 1995", title = "Scientific computing and validated numerics: proceedings of the International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics {SCAN}-95 held in Wuppertal, Germany, September 26--29, 1995", volume = "90", publisher = "Akademie Verlag", address = "Berlin, Germany", pages = "340", year = "1996", ISBN = "3-05-501737-4", ISBN-13 = "978-3-05-501737-7", ISSN = "0138-3019", LCCN = "QA76.95 .I575 1995", bibdate = "Fri Mar 27 09:56:17 MST 1998", bibsource = "https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib", series = "Mathematical Research", acknowledgement = ack-nhfb, } @Proceedings{Broy:1996:DPD, editor = "M. Broy", booktitle = "{Deductive program design: Proceedings of the NATO Advanced Study Institute on Deductive Program Design, held in Marktoberdorf, Germany, July 26--August 7, 1994}", title = "{Deductive program design: Proceedings of the NATO Advanced Study Institute on Deductive Program Design, held in Marktoberdorf, Germany, July 26--August 7, 1994}", volume = "152", publisher = pub-SV, address = pub-SV:adr, pages = "viii + 467", year = "1996", ISBN = "3-540-60947-4 (hardcover)", ISBN-13 = "978-3-540-60947-6 (hardcover)", LCCN = "QA76.9.D5 D38 1996", bibdate = "Tue Mar 17 10:33:47 MDT 2015", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/d/dijkstra-edsger-w.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; z3950.loc.gov:7090/Voyager", series = "NATO ASI series. Series F, Computer and systems sciences", URL = "http://www.loc.gov/catdir/enhancements/fy0812/96010788-d.html", acknowledgement = ack-nhfb, subject = "Electronic data processing; Distributed processing; Congresses; System design; Logic, Symbolic and mathematical", } @Book{Zhang:1996:AMI, editor = "H. (Hantao) Zhang", booktitle = "Automated Mathematical Induction", title = "Automated Mathematical Induction", publisher = pub-SV, address = pub-SV:adr, pages = "224", year = "1996", DOI = "https://doi.org/10.1007/978-94-009-1675-3", ISBN = "94-010-7250-7 (print), 94-009-1675-2 (e-book)", ISBN-13 = "978-94-010-7250-2 (print), 978-94-009-1675-3 (e-book)", LCCN = "Q334-342", bibdate = "Tue Mar 14 12:17:31 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://public.eblib.com/choice/publicfullrecord.aspx?p=3102529", abstract = "Two decades ago, Boyer and Moore built one of the first automated theorem provers that was capable of proofs by mathematical induction. Today, the Boyer--Moore theorem prover remains the most successful in the field. For a long time, the research on automated mathematical induction was confined to very few people. In recent years, as more people realize the importance of automated inductive reasoning to the use of formal methods of software and hardware development, more automated inductive proof systems have been built.\par Three years ago, the interested researchers in the field formed two consortia on automated inductive reasoning --- the MInd consortium in Europe and the IndUS consortium in the United States. The two consortia organized three joint workshops in 1992--1995. There will be another one in 1996. Following the suggestions of Alan Bundy and Deepak Kapur, this book documents advances in the understanding of the field and in the power of the theorem provers that can be built.\par In the first of six papers, the reader is provided with a tutorial study of the Boyer--Moore theorem prover. The other five papers present novel ideas that could be used to build theorem provers more powerful than the Boyer-Moore prover.", acknowledgement = ack-nhfb, subject = "Computer science; Logic; Artificial intelligence; Logic, Symbolic and mathematical; Artificial intelligence; Computer science; Logic; Logic, Symbolic and mathematical", tableofcontents = "Induction Using Term Orders \\ New Uses of Linear Arithmetic in Automated Theorem Proving by Induction \\ Productive Use of Failure in Inductive Proof \\ Middle-Out Reasoning for Synthesis and Induction \\ A Calculus for and Termination of Rippling \\ Interaction with the Boyer Moore Theorem Prover: A Tutorial Study Using the Arithmetic--Geometric Mean Theorem", } @Book{Berggren:1997:PSB, editor = "Lennart Berggren and Jonathan M. Borwein and Peter B. Borwein", booktitle = "Pi, a source book", title = "Pi, a source book", publisher = pub-SV, address = pub-SV:adr, pages = "xix + 716", year = "1997", DOI = "https://doi.org/10.1007/978-1-4757-2736-4", ISBN = "0-387-94924-0, 1-4757-2736-4 (e-book), 1-4757-2738-0 (print), 3-540-94924-0", ISBN-13 = "978-0-387-94924-6, 978-1-4757-2736-4 (e-book), 978-1-4757-2738-8 (print), 978-3-540-94924-4", LCCN = "QA484 .P5 1997", bibdate = "Fri Sep 2 17:41:50 MDT 2022", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib; z3950.loc.gov:7090/Voyager", abstract = "The aim of this book is to provide a complete history of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics, and a surprising amount of the most important mathematics and mathematicians have contributed to its unfolding. Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. Yet, despite this, no source book on pi has been published. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet still accessible, mathematics. Mathematicians and historians of mathematics will find this book indispensable. Teachers at every level from the seventh grade onward will find here ample resources for anything from special topic courses to individual talks and special student projects. The literature on pi included in this source book falls into three classes: first a selection of the mathematical literature of four millennia, second a variety of historical studies or writings on the cultural meaning and significance of the number, and third, a number of treatments on pi that are fanciful, satirical and/or whimsical.", acknowledgement = ack-nhfb, ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject = "Pi; Pi (Le nombre); Pi.; Pi (le nombre)", tableofcontents = "Preface / v \\ \\ Acknowledgments / ix \\ \\ Introduction / xvii \\ \\ 1. The Rhind Mathematical Papyrus-Problem 50 ($\approx$ 1650 B.C.) / A problem dealing with the area of a round field of given diameter / 1 \\ \\ 2. Engels. Quadrature of the Circle in Ancient Egypt (1977) / A conjectural explanation of how the mathematicians of ancient Egypt approximated the area of a circle / 3 \\ \\ 3. Archimedes. Measurement of a Circle ($\approx$ 250 BC) / The seminal work in which Archimedes presents the first true algorithm for $\pi$ / 7 \\ \\ 4. Phillips. Archimedes the Numerical Analyst (1981) / A summary of Archimedes' work on the computation of $\pi$ using modern notation / 15 \\ \\ 5. Lam and Ang. Circle Measurements in Ancient China (1986) / This paper discusses and contains a translation of Liu Hui's (3rd century) method for evaluating $\pi$ and also examines values for $\pi$ given by Zu Chongzhi (429--500) / 20 \\ \\ 6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane and Solid Figures ($\approx$ 850) / This extract gives an explicit statement and proof that the ratio of the circumference to the diameter is constant / 36 \\ \\ 7. M{\=a}dhava. The Power Series for Arctan and Pi ($\approx$ 1400) / These theorems by a fifteenth century Indian mathematician give Gregory's series for arctan with remainder terms and Leibniz's series for $\pi$ / 45 \\ \\ 8. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) / Correspondence about van Ceulen's tombstone in reference to it containing some digits of $\pi$ / 51 \\ \\ 9. Vi{\'e}te. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) / Two excerpts. One containing the first infinite expression of $\pi$, obtained by relating the area of a regular $2n$-gon to that of a regular $n$-gon / 53 \\ \\ 10. Wallis. Computation of $\pi$ by Successive Interpolations (1655) / How Wallis derived the infinite product for $\pi$ that bears his name / 68 \\ \\ 11. Wallis. Arithmetica Infinitorum (1655) / An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm. Brounker's continued fraction expression for $4/\pi$ / 78 \\ \\ 12. Huygens. De Circuli Magnitudine Inventa (1724) / Huygens's proof of W. Snell's discovery of improvements in Archimedes' method of estimating the lengths of circular arcs / 81 \\ \\ 13. Gregory. Correspondence with John Collins (1671) / A letter to Collins in which he gives his series for arctangent, carried to the ninth power. / 87 \\ \\ 14. Roy. The Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha (1990) / A discussion of the discovery of the series $\pi/4 = 1 - 1/3 + 1/5, \cdots{}$ / 92 \\ \\ 15. Jones. The First Use of $\pi$ for the Circle Ratio (1706) / An excerpt from Jones' book, the Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics, London, 1706 / 108 \\ \\ 16. Newton. Of the Method of Fluxions and Infinite Series (1737) / An excerpt giving Newton's calculation of $\pi$ to 16 decimal places / 110 \\ \\ 17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748) / This includes many of Euler's infinite series for $\pi$ and powers of $\pi$ / 112 \\ \\ 18. Lambert. M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques (1761) / An excerpt from Lambert's original proof of the irrationality of $\pi$ / 129 \\ \\ 19. Lambert. Irrationality of $\pi$ (1969) / A translation and Struik's discussion of Lambert's proof of the irrationality of $\pi$ / 141 \\ \\ 20. Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) / Pages from Shank's report of his monumental hand calculation of $\pi$ / 147 \\ \\ 21. Hermite. Sur La Fonction Exponentielle (1873) / The first proof of the transcendence of $e$ / 162 \\ \\ 22. Lindemann. Ueber die Zahl $\pi$ (1882) / The first proof of the transcendence of $\pi$ / 194 \\ \\ 23. Weierstrass. Zu Lindemann's Abhandlung ``Uber die Ludolphsche Zahl'' (1885) / Weierstrass' proof of the transcendence of $\pi$ / 207 \\ \\ 24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und $\pi$ (1893) / Hilbert's short and elegant simplification of the transcendence proofs for $e$ and $\pi$ / 226 \\ \\ 25. Goodwin. Quadrature of the Circle (1894) / The dubious origin of the attempted legislation of the value of $\pi$ in Indiana / 230 \\ \\ 26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) / A summary of the action taken by the Indiana State Legislature to fix the value of $\pi$ (including a copy of the actual bill that was proposed) / 231 \\ \\ 27. Singmaster. The Legal Values of Pi (1985) / A history of the attempt by Indiana to legislate the value of $\pi$ / 236 \\ \\ 28. Ramanujan. Squaring the Circle (1913) / A geometric approximation to $\pi$ / 240 \\ \\ 29. Ramanujan. Modular Equations and Approximations to $\pi$ (1914) / Ramanujan's seminal paper on $\pi$ that includes a number of striking series and algebraic approximations / 241 \\ \\ 30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) / A Presidential address to the Mathematical Association in which the author gives an account of ``some of the elementary work on arcs and ellipses and other curves which led up to the idea of inverting an elliptic integral, and so laying the foundations of elliptic functions and doubly periodic functions generally.'' / 258 \\ \\ 31. Ballantine. The Best (?) Formula for Computing $\pi$ to a Thousand Places (1939) / An early attempt to orchestrate the calculation of $\pi$ more cleverly / 271 \\ \\ 32. Birch. An Algorithm for Construction of Arctangent Relations (1946) / The object of this note is to express $\pi / 4 $ as a sum of arctan relations in powers of 10 / 274 \\ \\ 33. Niven. A Simple Proof that $\pi$ Is Irrational (1947) / A very concise proof of the irrationality of $\pi$ / 276 \\ \\ 34. Reitwiesner. An ENIAC Determination of $\pi$ and $e$ to 2000 Decimal Places (1950) / One of the first computer-based computations / 277 \\ \\ 35. Schepler. The Chronology of Pi (1950) / A fairly reliable outline of the history of $\pi$ from 3000 BC to 1949 / 282 \\ \\ 36. Mahler. On the Approximation of $\pi$ (1953) / ``The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of $\pi$ from a given rational or algebraic number'' / 306 \\ \\ 37. Wrench, Jr. The Evolution of Extended Decimal Approximations to $\pi$ (1960) / A history of the calculation of the digits of $\pi$ to 1960 \\ \\ 38. Shanks and Wrench, Jr. Calculation of $\pi$ to 100,000 Decimals (1962) / A landmark computation of $\pi$ to more than 100,000 places / 326 \\ \\ 39. Sweeny. On the Computation of Euler's Constant (1963) / The computation of Euler's constant to 3566 decimal places / 350 \\ \\ 40. Baker. Approximations to the Logarithms of Certain Rational Numbers (1964) / The main purpose of this deep and fundamental paper is to ``deduce results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.'' / 359 \\ \\ 41. Adams. Asymptotic Diophantine Approximations to $E$ (1966) / An asymptotic estimate for the rational approximation to $e$ which disproves the conjecture that $e$ behaves like almost all numbers in this respect / 368 \\ \\ 42. Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) / An important extension of Hilbert's approach to the study of transcendence / 372 \\ \\ 43. Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) / A collection of mathematical stories and anecdotes about $\pi$ / 400 \\ \\ 44. Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) / A further collection of mathematical stories and anecdotes about $\pi$ / 402 \\ \\ 45. Todd. The Lemniscate Constants (1975) / A unifying account of some of the methods used for computing the lemniscate constants / 412 \\ \\ 46. Salamin. Computation of r Using Arithmetic-Geometric Mean (1976) / The first quadratically converging algorithm for $\pi$ based on Gauss's AGM and on Legendre's relation for elliptic integrals / 418 \\ \\ 47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) / ``This paper contains the `Gauss-Legendre' method and some different algorithms for log and exp (using Landen transformations).'' / 424 \\ \\ 48. Beukers. A Note on the Irrationality of $\zeta(2)$ and $\zetq(3)$ (1979) / A short and elegant recasting of Ap{\'e}ry's proof of the irrationality of $\zeta(3)$ (and $\zeta(2)$) / 434 \\ \\ 49. van der Poorten. A Proof that Euler Missed \ldots{} Ap{\'e}ry's Proof of the Irrationality of $\zeta(3)$ (1979) / An illuminating account of Ap{\'e}ry's astonishing proof of the irrationality of $\zeta(3)$ / 439 \\ \\ 50. Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980) / Several new algorithms for high precision calculation of Euler's constant, including one which was used to compute 30,100 decimal places / 448 \\ \\ 51. Apostol. A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) / This note shows that one of the double integrals considered by Beukers ([48] in the table of contents) can be used to establish directly that $\zeta(2) = \pi / 6$ / 456 \\ \\ 52. O'Shaughnessy. Putting God Back in Math (1983) / An article about the Institute of Pi Research, an organization that ``pokes fun at creationists by pointing out that even the Bible makes mistakes.'' / 458 \\ \\ 53. Stern. A Remarkable Approximation to $\pi$ (1985) / Justification of the value of $\pi$ in the Bible through numerological interpretations / 460 \\ \\ 54. Newman and Shanks. On a Sequence Arising in Series for $\pi$ (1984) / More connections between $\pi$ and modular equations / 462 \\ \\ 55. Cox. The Arithmetic-Geometric Mean of Gauss (1984) / An extensive study of the complex analytic properties of the AGM / 481 \\ \\ 56. Borwein and Borwein. The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions (1984) / The relationship between the AGM iteration and fast computation of elementary functions (one of the by-products is an algorithm for $\pi$) / 537 \\ \\ 57. Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) / Elementary algorithms for evaluating $e^x$ and $\pi$ using the Gauss AGM without explicit elliptic function theory / 553 \\ \\ 58. Wagon. Is Pi Normal? (1985) / A discussion of the conjecture that $\pi$ has randomly distributed digits / 557 \\ \\ 59. Keith. Circle Digits: A Self-Referential Story (1986) / A mnemonic for the first 402 decimal places of $\pi$ / 560 \\ \\ 60. Bailey. The Computation of $\pi$ to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988) / The algorithms used, both for $\pi$ and for performing the required multiple-precision arithmetic / 562 \\ \\ 61. Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of 1 Calculation (1988) / Details of the computation and statistical tests of the first 200 million digits of $\pi$ / 576 \\ \\ 62. Borwein and Borwein. Ramanujan and Pi (1988) / This article documents Ramanujan's life, his ingenious approach to calculating $\pi$, and how his approach is now incorporated into modern computer algorithms / 588 \\ \\ 63. Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988) / This excerpt describes ``Ramanujan's original quadratic period--quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of $\pi$ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series.'' / 596 \\ \\ 64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) / An exposition of the computation of $\pi$ using mathematics rooted in Ramanujan's work / 623 \\ \\ 65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989) / An explanation as to why the slowly convergent Gregory series for $\pi$, truncated at 500,000 terms, gives $\pi$ to 40 places with only the 6th, 17th, 18th, and 29th places being incorrect / 642 \\ \\ 66. Beukers, B{\'e}zivin, and Robba. An Alternative Proof of the Lindemann--Weierstrass Theorem (1990) / The Lindemann--Weierstrass theorem as a by-product of a criterion for rationality of solutions of differential equations / 649 \\ \\ 67. Webster. The Tail of Pi (1991) / Various anecdotes about $\pi$ from the 14th annual IMO Lecture to the Royal Society / 654 \\ \\ 68. Eco. An excerpt from Foucault's Pendulum (1993) / ``The unnumbered perfection of the circle itself.'' / 658 \\ \\ 69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996) / A mnemonic for $\pi$ based on Edgar Allen Poe's poem ``The Raven.'' / 659 \\ \\ 70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996) / A fast method for computing individual digits of $\pi$ in base 2 / 663 \\ Appendix I --- On the Early History of Pi / 677 \\ \\ Appendix II --- A Computational Chronology of Pi / 683 \\ \\ Appendix III --- Selected Formulae for Pi / 686 \\ \\ Bibliography / 690 \\ \\ Credits / 697 \\ \\ Index / 701", } @Book{Berggren:2000:PSB, editor = "Lennart Berggren and Jonathan Borwein and Peter Borwein", booktitle = "Pi: a source book", title = "Pi: a source book", publisher = pub-SV, address = pub-SV:adr, edition = "Second", pages = "xx + 736", year = "2000", DOI = "https://doi.org/10.1007/978-1-4757-3240-5", ISBN = "0-387-98946-3 (hardcover)", ISBN-13 = "978-0-387-98946-4 (hardcover)", LCCN = "QA484 .P5 2000", MRclass = "11-00 (01A05 01A75 11-03)", MRnumber = "1746004", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", libnote = "Not yet in my library.", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", subject = "Pi (mathematical constant)", tableofcontents = "Preface / v \\ \\ Preface to the Second Edition / viii \\ Acknowledgments / ix \\ \\ Introduction / xvii \\ \\ 1. The Rhind Mathematical Papyrus-Problem 50 ($\approx$ 1650 B.C.) / A problem dealing with the area of a round field of given diameter / 1 \\ \\ 2. Engels. Quadrature of the Circle in Ancient Egypt (1977) / A conjectural explanation of how the mathematicians of ancient Egypt approximated the area of a circle / 3 \\ \\ 3. Archimedes. Measurement of a Circle ($\approx$ 250 BC) / The seminal work in which Archimedes presents the first true algorithm for $\pi$ / 7 \\ \\ 4. Phillips. Archimedes the Numerical Analyst (1981) / A summary of Archimedes' work on the computation of $\pi$ using modern notation / 15 \\ \\ 5. Lam and Ang. Circle Measurements in Ancient China (1986) / This paper discusses and contains a translation of Liu Hui's (3rd century) method for evaluating $\pi$ and also examines values for $\pi$ given by Zu Chongzhi (429--500) / 20 \\ \\ 6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane and Solid Figures ($\approx$ 850) / This extract gives an explicit statement and proof that the ratio of the circumference to the diameter is constant / 36 \\ \\ 7. M{\=a}dhava. The Power Series for Arctan and Pi ($\approx$ 1400) / These theorems by a fifteenth century Indian mathematician give Gregory's series for arctan with remainder terms and Leibniz's series for $\pi$ / 45 \\ \\ 8. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) / Correspondence about van Ceulen's tombstone in reference to it containing some digits of $\pi$ / 51 \\ \\ 9. Vi{\'e}te. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) / Two excerpts. One containing the first infinite expression of $\pi$, obtained by relating the area of a regular $2n$-gon to that of a regular $n$-gon / 53 \\ \\ 10. Wallis. Computation of $\pi$ by Successive Interpolations (1655) / How Wallis derived the infinite product for $\pi$ that bears his name / 68 \\ \\ 11. Wallis. Arithmetica Infinitorum (1655) / An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm. Brounker's continued fraction expression for $4/\pi$ / 78 \\ \\ 12. Huygens. De Circuli Magnitudine Inventa (1724) / Huygens's proof of W. Snell's discovery of improvements in Archimedes' method of estimating the lengths of circular arcs / 81 \\ \\ 13. Gregory. Correspondence with John Collins (1671) / A letter to Collins in which he gives his series for arctangent, carried to the ninth power. / 87 \\ \\ 14. Roy. The Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha (1990) / A discussion of the discovery of the series $\pi/4 = 1 - 1/3 + 1/5, \cdots{}$ / 92 \\ \\ 15. Jones. The First Use of $\pi$ for the Circle Ratio (1706) / An excerpt from Jones' book, the Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics, London, 1706 / 108 \\ \\ 16. Newton. Of the Method of Fluxions and Infinite Series (1737) / An excerpt giving Newton's calculation of $\pi$ to 16 decimal places / 110 \\ \\ 17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748) / This includes many of Euler's infinite series for $\pi$ and powers of $\pi$ / 112 \\ \\ 18. Lambert. M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques (1761) / An excerpt from Lambert's original proof of the irrationality of $\pi$ / 129 \\ \\ 19. Lambert. Irrationality of $\pi$ (1969) / A translation and Struik's discussion of Lambert's proof of the irrationality of $\pi$ / 141 \\ \\ 20. Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) / Pages from Shank's report of his monumental hand calculation of $\pi$ / 147 \\ \\ 21. Hermite. Sur La Fonction Exponentielle (1873) / The first proof of the transcendence of $e$ / 162 \\ \\ 22. Lindemann. Ueber die Zahl $\pi$ (1882) / The first proof of the transcendence of $\pi$ / 194 \\ \\ 23. Weierstrass. Zu Lindemann's Abhandlung ``Uber die Ludolphsche Zahl'' (1885) / Weierstrass' proof of the transcendence of $\pi$ / 207 \\ \\ 24. Hilbert. Ueber die Trancendenz der Zahlen $e$ und $\pi$ (1893) / Hilbert's short and elegant simplification of the transcendence proofs for $e$ and $\pi$ / 226 \\ \\ 25. Goodwin. Quadrature of the Circle (1894) / The dubious origin of the attempted legislation of the value of $\pi$ in Indiana / 230 \\ \\ 26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) / A summary of the action taken by the Indiana State Legislature to fix the value of $\pi$ (including a copy of the actual bill that was proposed) / 231 \\ \\ 27. Singmaster. The Legal Values of Pi (1985) / A history of the attempt by Indiana to legislate the value of $\pi$ / 236 \\ \\ 28. Ramanujan. Squaring the Circle (1913) / A geometric approximation to $\pi$ / 240 \\ \\ 29. Ramanujan. Modular Equations and Approximations to $\pi$ (1914) / Ramanujan's seminal paper on $\pi$ that includes a number of striking series and algebraic approximations / 241 \\ \\ 30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) / A Presidential address to the Mathematical Association in which the author gives an account of ``some of the elementary work on arcs and ellipses and other curves which led up to the idea of inverting an elliptic integral, and so laying the foundations of elliptic functions and doubly periodic functions generally.'' / 258 \\ \\ 31. Ballantine. The Best (?) Formula for Computing $\pi$ to a Thousand Places (1939) / An early attempt to orchestrate the calculation of $\pi$ more cleverly / 271 \\ \\ 32. Birch. An Algorithm for Construction of Arctangent Relations (1946) / The object of this note is to express $\pi / 4 $ as a sum of arctan relations in powers of 10 / 274 \\ \\ 33. Niven. A Simple Proof that $\pi$ Is Irrational (1947) / A very concise proof of the irrationality of $\pi$ / 276 \\ \\ 34. Reitwiesner. An ENIAC Determination of $\pi$ and $e$ to 2000 Decimal Places (1950) / One of the first computer-based computations / 277 \\ \\ 35. Schepler. The Chronology of Pi (1950) / A fairly reliable outline of the history of $\pi$ from 3000 BC to 1949 / 282 \\ \\ 36. Mahler. On the Approximation of $\pi$ (1953) / ``The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of $\pi$ from a given rational or algebraic number'' / 306 \\ \\ 37. Wrench, Jr. The Evolution of Extended Decimal Approximations to $\pi$ (1960) / A history of the calculation of the digits of $\pi$ to 1960 \\ \\ 38. Shanks and Wrench, Jr. Calculation of $\pi$ to 100,000 Decimals (1962) / A landmark computation of $\pi$ to more than 100,000 places / 326 \\ \\ 39. Sweeny. On the Computation of Euler's Constant (1963) / The computation of Euler's constant to 3566 decimal places / 350 \\ \\ 40. Baker. Approximations to the Logarithms of Certain Rational Numbers (1964) / The main purpose of this deep and fundamental paper is to ``deduce results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.'' / 359 \\ \\ 41. Adams. Asymptotic Diophantine Approximations to $E$ (1966) / An asymptotic estimate for the rational approximation to $e$ which disproves the conjecture that $e$ behaves like almost all numbers in this respect / 368 \\ \\ 42. Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) / An important extension of Hilbert's approach to the study of transcendence / 372 \\ \\ 43. Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) / A collection of mathematical stories and anecdotes about $\pi$ / 400 \\ \\ 44. Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) / A further collection of mathematical stories and anecdotes about $\pi$ / 402 \\ \\ 45. Todd. The Lemniscate Constants (1975) / A unifying account of some of the methods used for computing the lemniscate constants / 412 \\ \\ 46. Salamin. Computation of r Using Arithmetic-Geometric Mean (1976) / The first quadratically converging algorithm for $\pi$ based on Gauss's AGM and on Legendre's relation for elliptic integrals / 418 \\ \\ 47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) / ``This paper contains the `Gauss-Legendre' method and some different algorithms for log and exp (using Landen transformations).'' / 424 \\ \\ 48. Beukers. A Note on the Irrationality of $\zeta(2)$ and $\zetq(3)$ (1979) / A short and elegant recasting of Ap{\'e}ry's proof of the irrationality of $\zeta(3)$ (and $\zeta(2)$) / 434 \\ \\ 49. van der Poorten. A Proof that Euler Missed \ldots{} Ap{\'e}ry's Proof of the Irrationality of $\zeta(3)$ (1979) / An illuminating account of Ap{\'e}ry's astonishing proof of the irrationality of $\zeta(3)$ / 439 \\ \\ 50. Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980) / Several new algorithms for high precision calculation of Euler's constant, including one which was used to compute 30,100 decimal places / 448 \\ \\ 51. Apostol. A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) / This note shows that one of the double integrals considered by Beukers ([48] in the table of contents) can be used to establish directly that $\zeta(2) = \pi / 6$ / 456 \\ \\ 52. O'Shaughnessy. Putting God Back in Math (1983) / An article about the Institute of Pi Research, an organization that ``pokes fun at creationists by pointing out that even the Bible makes mistakes.'' / 458 \\ \\ 53. Stern. A Remarkable Approximation to $\pi$ (1985) / Justification of the value of $\pi$ in the Bible through numerological interpretations / 460 \\ \\ 54. Newman and Shanks. On a Sequence Arising in Series for $\pi$ (1984) / More connections between $\pi$ and modular equations / 462 \\ \\ 55. Cox. The Arithmetic-Geometric Mean of Gauss (1984) / An extensive study of the complex analytic properties of the AGM / 481 \\ \\ 56. Borwein and Borwein. The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions (1984) / The relationship between the AGM iteration and fast computation of elementary functions (one of the by-products is an algorithm for $\pi$) / 537 \\ \\ 57. Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) / Elementary algorithms for evaluating $e^x$ and $\pi$ using the Gauss AGM without explicit elliptic function theory / 553 \\ \\ 58. Wagon. Is Pi Normal? (1985) / A discussion of the conjecture that $\pi$ has randomly distributed digits / 557 \\ \\ 59. Keith. Circle Digits: A Self-Referential Story (1986) / A mnemonic for the first 402 decimal places of $\pi$ / 560 \\ \\ 60. Bailey. The Computation of $\pi$ to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988) / The algorithms used, both for $\pi$ and for performing the required multiple-precision arithmetic / 562 \\ \\ 61. Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of 1 Calculation (1988) / Details of the computation and statistical tests of the first 200 million digits of $\pi$ / 576 \\ \\ 62. Borwein and Borwein. Ramanujan and Pi (1988) / This article documents Ramanujan's life, his ingenious approach to calculating $\pi$, and how his approach is now incorporated into modern computer algorithms / 588 \\ \\ 63. Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988) / This excerpt describes ``Ramanujan's original quadratic period--quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of $\pi$ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series.'' / 596 \\ \\ 64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) / An exposition of the computation of $\pi$ using mathematics rooted in Ramanujan's work / 623 \\ \\ 65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989) / An explanation as to why the slowly convergent Gregory series for $\pi$, truncated at 500,000 terms, gives $\pi$ to 40 places with only the 6th, 17th, 18th, and 29th places being incorrect / 642 \\ \\ 66. Beukers, B{\'e}zivin, and Robba. An Alternative Proof of the Lindemann--Weierstrass Theorem (1990) / The Lindemann--Weierstrass theorem as a by-product of a criterion for rationality of solutions of differential equations / 649 \\ \\ 67. Webster. The Tail of Pi (1991) / Various anecdotes about $\pi$ from the 14th annual IMO Lecture to the Royal Society / 654 \\ \\ 68. Eco. An excerpt from Foucault's Pendulum (1993) / ``The unnumbered perfection of the circle itself.'' / 658 \\ \\ 69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996) / A mnemonic for $\pi$ based on Edgar Allen Poe's poem ``The Raven.'' / 659 \\ \\ 70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996) / A fast method for computing individual digits of $\pi$ in base 2 / 663 \\ Appendix I --- On the Early History of Pi / 677 \\ \\ Appendix II --- A Computational Chronology of Pi / 683 \\ \\ Appendix III --- Selected Formulae for Pi / 686 \\ \\ Appendix IV --- Translations of Vi{\`e}te and Huygens / 690 \\ Bibliography / 711 \\ \\ Credits / 717 \\ \\ Index / 721", } @Book{Berggren:2004:PSB, editor = "Lennart Berggren and Jonathan Borwein and Peter Borwein", booktitle = "Pi: a source book", title = "Pi: a source book", publisher = pub-SV, address = pub-SV:adr, edition = "Third", pages = "xx + 797", year = "2004", DOI = "https://doi.org/10.1007/978-1-4757-4217-6", ISBN = "0-387-20571-3", ISBN-13 = "978-0-387-20571-7", MRclass = "11-00 (01A05 01A75 11-03)", MRnumber = "2065455", MRreviewer = "F. Beukers", bibdate = "Wed Aug 10 11:09:47 2016", bibsource = "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib", acknowledgement = ack-nhfb, author-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646", remark = "CECM Preprint 2003:210.", tableofcontents = "Preface to the Third Edition / v \\ Preface to the Second Edition / vi \\ Preface / vii \\ Acknowledgments / x \\ Introduction / xvii \\ 1. The Rhind Mathematical Papyrus --- Problem 50 ($\approx$1650 B.C.) / A problem dealing with the area of a round field of given diameter / 1 \\ 2. Engels / Quadrature of the Circle in Ancient Egypt (1977) / A conjectural explanation of how the mathematicians of ancient Egypt approximated the area of a circle / 3 \\ 3. Archimedes / Measurement of a Circle --- (-250 B.C.) / The seminal work in which Archimedes presents the first true algorithm for $ \pi $ / 7 \\ 4. Phillips / Archimedes the Numerical --- Analyst (1981) / A summary of Archimedes' work on the computation of $ \pi $ using modem notation / 15 \\ 5. Lam and Ang / Circle Measurements in Ancient China (1986) / This paper discusses and contains a translation of Liu Hui's (3rd century) method for evaluating $ \pi $ and also examines values for $ \pi $ given by Zu Chongzhi (429--500) / 20 \\ 6. The Ban{\=u} M{\=u}s{\=a}: The Measurement of Plane and Solid Figures (--850) / This extract gives an explicit statement and proof that the ratio of the circumference to the diameter is constant / 36 \\ 7. M{\=a}dhava / The Power Series for Arctan and Pi (-1400) / These theorems by a fifteenth century Indian mathematician give Gregory's series for arctan with remainder terms and Leibniz's series for $ \pi $ / 45 \\ 8. Hope-Jones / Ludolph (or Ludolff or Lucius) van Ceulen (1938) / Correspondence about van Ceulen's tombstone in reference to it containing some digits of $ \pi $ / 51 \\ 9. Vi{\`e}te / \booktitle{Variorum de Rebus Mathematicis Reponsorum Liber VII} (1593) / Two excerpts. One containing the first infinite expression of $ \pi $, obtained by relating the area of a regular $2n$-gon to that of a regular $n$-gon / 53 \\ 10. Wallis. Computation of $ \pi $ by Successive Interpolations (1655) / How Wallis derived the infinite product for $ \pi $ that bears his name / 68 \\ 11. Wallis / \booktitle{Arithmetica Infinitorum} (1655) / An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm. Brounker's continued fraction expression for $ 4 / \pi$ / ?? \\ 12. Huygens / \booktitle{De Circuli Magnitudine Inventa} (1654) / Huygens's demonstration of how to triple the number of correct decimals over those in Archimedes' estimate of $ \pi $ / 81 13. Gregory / Correspondence with John Collins (1671) / A letter to Collins in which he gives his series for arctangent, carried to the ninth power / 87 \\ 14. Roy / The Discovery of the Series Formula for $ \pi $ by Leibniz, Gregory, and Nilakantha (1990) / A discussion of the discovery of the series $ \pi / 4 = 1 - 1/3 + 1/5 - \cdots{} $ / 92 \\ 15. Jones / The First Use of $ \pi $ for the Circle Ratio (1706) / An excerpt from Jones' book, the \booktitle{Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics}, London, 1706 / 108 \\ 16. Newton / Of the Method of Fluxions and Infinite Series (1737) / An excerpt giving Newton's calculation of $ \pi $ to 16 decimal places / 110 \\ 17. Euler / Chapter 10 of \booktitle{Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series)} (1748) / This includes many of Euler's infinite series for $ \pi $ and powers of $ \pi $ / 112 \\ 18. Lambert / \booktitle{M{\'e}moire Sur Quelques Propri{\'e}t{\'e}s Remarquables Des Quantit{\'e}s Transcendentes Circulaires et Logarithmiques} (1761) / An excerpt from Lambert's original proof of the irrationality of $ \pi $ / 129 19. Lambert / Irrationality of $ \pi $ (1969) / A translation and Struik's discussion of Lambert's proof of the irrationality of $ \pi $ / 141 \\ 20. Shanks / Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) / Pages from Shanks's report of his monumental hand calculation of $ \pi $ / 147 \\ 21. Hermite / \booktitle{Sur La Fonction Exponentielle} (1873) / The first proof of the transcendence of $ e $ / 162 \\ 22. Lindemann / \booktitle{Ueber die Zahl $ \pi $} (1882) / The first proof of the transcendence of $ \pi $ / 194 23. Weierstrass / \booktitle{Zu Lindemann's Abhandlung ``{\"U}ber die Ludolphsche Zahl''} (1885) / Weierstrass' proof of the transcendence of $ \pi $ / 207 24. Hilbert / \booktitle{Ueber die Transzendenz der Zahlen $ e $ und $ \pi $} (1893) / Hilbert's short and elegant simplification of the transcendence proofs for $ e $ and $ \pi $ / 226 25. Goodwin / Quadrature of the Circle (1894) / The dubious origin of the attempted legislation of the value of $ \pi $ in Indiana / 230 \\ 26. Edington / House Bill No. 246, Indiana State Legislature, 1897 (1935) / A summary of the action taken by the Indiana State Legislature to fix the value of $ \pi $ (including a copy of the actual bill that was proposed) / 231 \\ 27. Singmaster / The Legal Values of Pi (1985) / A history of the attempt by Indiana to legislate the value of $ \pi $ / 236 \\ 28. Ramanujan / Squaring the Circle (1913) / A geometric approximation to $ \pi $ / 240 \\ 29. Ramanujan / Modular Equations and Approximations to $ \pi $ (1914) / Ramanujan's seminal paper on pi that includes a number of striking series and algebraic approximations / 241 \\ 30. Watson / The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) / A Presidential address to the Mathematical Association in which the author gives an account of ``some of the elementary work on arcs and ellipses and other curves which led up to the idea of inverting an elliptic integral, and so laying the foundations of elliptic functions and doubly periodic functions generally.'' / ?? \\ 31. Ballantine / The Best (?) Formula for Computing $ \pi $ to a Thousand Places (1939) / An early attempt to orchestrate the calculation of $ \pi $ more cleverly / 271 \\ 32. Birch / An Algorithm for Construction of Arctangent Relations (1946) / The object of this note is to express $ \pi / 4$ as a sum of arctan relations in powers of 10 / 274 \\ 33. Niven / A Simple Proof that $ \pi $ is Irrational (1947) / A very concise proof of the irrationality of $ \pi $ / 276 \\ 34. Reitwiesner / An ENIAC Determination of $ \pi $ and $ e $ to 2000 Decimal Places (1950) / One of the first computer-based computations / 277 \\ 35. Schepler / The Chronology of Pi (1950) / A fairly reliable outline of the history of $ \pi $ from 3000 B.C. to 1949 / 282 \\ 36. Mahler / On the Approximation of $ \pi $ (1953) / ``The aim of this paper is to determine an explicit lower bound free of unknown constants for the distance of $ \pi $ from a given rational or algebraic number.'' / 306 \\ 37. Wrench, Jr. / The Evolution of Extended Decimal Approximations to $ \pi $ (1960) / A history of the calculation of the digits of $ \pi $ to 1960 / 319 \\ 38. Shanks and Wrench, Jr. / Calculation of $ \pi $ to 100,000 Decimals (1962) / A landmark computation of $ \pi $ to more than 100,000 places / 326 39. Sweeny / On the Computation of Euler's Constant (1963) / The computation of Euler's constant to 3566 decimal places / 350 40. Baker / Approximations to the Logarithms of Certain Rational Numbers (1964) / The main purpose of this deep and fundamental paper is to ``deduce results concerning the accuracy with which the natural logarithms of certain rational numbers may be approximated by rational numbers, or, more generally, by algebraic numbers of bounded degree.'' / 359 \\ 41. Adams / Asymptotic Diophantine Approximations to e (1966) / An asymptotic estimate for the rational approximation to $ e $ which disproves the conjecture that $ e $ behaves like almost all numbers in this respect / 368 \\ 42. Mahler / Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) / An important extension of Hilbert's approach to the study of transcendence / 372 43. Eves / In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) / A collection of mathematical stories and anecdotes about $ \pi $ / 456 \\ 44. Eves / Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) / A further collection of mathematical stories and anecdotes about $ \pi $ / 402 45. Todd / The Lemniscate Constants (1975) / A unifying account of some of the methods used for computing the lemniscate constants / 412 \\ 46. Salamin / Computation of $ \pi $ Using Arithmetic--Geometric Mean (1976) / The first quadratically converging algorithm for $ \pi $ based on Gauss's AGM and on Legendre's relation for elliptic integrals / 418 \\ 47. Brent / Fast Multiple-Precision Evaluation of Elementary Functions (1976) / ``This paper contains the `Gauss--Legendre' method and some different algorithms for $\log$ and $\exp$ (using Landen transformations).'' / 424 \\ 48. Beukers / A Note on the Irrationality of $ \zeta(2) $ and $ \zeta(3) $ (1979) / A short and elegant recasting of Apery's proof of the irrationality of $\zeta(3)$ (and $\zeta(2)$) / 434 \\ 49. van der Poorten / A Proof that Euler Missed \ldots{} Apery's Proof of the Irrationality of $\zeta (3)$ (1979) / An illuminating account of Apery's astonishing proof of the irrationality of $\zeta (3)$ / 439 \\ 50. Brent and McMillan / Some New Algorithms for High-Precision Computation of Euler's Constant (1980) / Several new algorithms for high-precision calculation of Euler's constant, including one which was used to compute 30,100 decimal places / 448 \\ 51. Apostol / A Proof that Euler Missed: Evaluating $\zeta(2)$ the Easy Way (1983) / This note shows that one of the double integrals considered by Beukers ([48] in the table of contents) can be used to establish directly that $\zeta(2) = \pi^2 / 6$ / 456 \\ 52. O'Shaughnessy / Putting God Back in Math (1983) / An article about the Institute of Pi Research, an organization that ``pokes fun at creationists by pointing out that even the Bible makes mistakes.'' / 458 \\ 53. Stern / A Remarkable Approximation to $ \pi $ (1985) / Justification of the value of $ \pi $ in the Bible through numerological interpretations / 460 \\ 54. Newman and Shanks / On a Sequence Arising in Series for $ \pi $ (1984) / More connections between $ \pi $ and modular equations / 462 \\ 55. Cox / The Arithmetic--Geometric Mean of Gauss (1984) / An extensive study of the complex analytic properties of the AGM / 481 \\ 56. Borwein and Borwein / The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions (1984) / The relationship between the AGM iteration and fast computation of elementary functions (one of the by-products is an algorithm for $ \pi $) / 537 57. Newman / A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) / Elementary algorithms for evaluating $ e^x $ and $ \pi $ using the Gauss AGM without explicit elliptic function theory / 553 \\ 58. Wagon / Is Pi Normal? (1985) / A discussion of the conjecture that $ \pi $ has randomly distributed digits / 557 \\ 59. Keith / Circle Digits: A Self-Referential Story (1986) / A mnemonic for the first 402 decimal places of $ \pi $ / 560 \\ 60. Bailey / The Computation of $ \pi $ to 29,360,000 Decimal Digits Using Borwein's Quartically Convergent Algorithm (1988) / The algorithms used, both for $ \pi $ and for performing the required multiple-precision arithmetic / 562 \\ 61. Kanada / Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of $ \pi $ Calculation (1988) / Details of the computation and statistical tests of the first 200 million digits of $ \pi $ / 576 \\ 62. Borwein and Borwein / Ramanujan and Pi (1988) / This article documents Ramanujan's life, his ingenious approach to calculating $ \pi $, and how his approach is now incorporated into modern computer algorithms / 588 \\ 63. Chudnovsky and Chudnovsky / Approximations and Complex Multiplication According to Ramanujan (1988) / This excerpt describes ``Ramanujan's original quadratic period--quasiperiod relations for elliptic curves with complex multiplication and their applications to representations of fractions of $ \pi $ and other logarithms in terms of rapidly convergent nearly integral (hypergeometric) series.'' / 596 \\ 64. Borwein, Borwein and Bailey / Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) / An exposition of the computation of $ \pi $ using mathematics rooted in Ramanujan's work / 623 \\ 65. Borwein, Borwein and Dilcher / Pi, Euler Numbers, and Asymptotic Expansions (1989) / An explanation as to why the slowly convergent Gregory series for $ \pi $, truncated at 500,000 terms, gives $ \pi $ to 40 places with only the 6th, 17th, 18th, and 29th places being incorrect / 642 \\ 66. Beukers, Bezivin, and Robba / An Alternative Proof of the Lindemann--Weierstrass Theorem (1990) / The Lindemann--Weierstrass theorem as a by-product of a criterion for rationality of solutions of differential equations / 649 \\ 67. Webster / The Tale of Pi (1991) / Various anecdotes about $ \pi $ from the 14th annual IMO Lecture to the Royal Society / 654 \\ 68. Eco / An excerpt from Foucault's Pendulum (1993) / ``The unnumbered perfection of the circle itself.'' / 658 \\ 69. Keith / Pi Mnemonics and the Art of Constrained Writing (1996) / A mnemonic for $ \pi $ based on Edgar Allen Poe's poem ``The Raven.'' / 659 \\ 70. Bailey, Borwein, and Plouffe / On the Rapid Computation of Various Polylogarithmic Constants (1997) / A fast method for computing individual digits of $ \pi $ in base 2 / 663 \\ Appendix I --- On the Early History of Pi / 677 \\ Appendix II --- A Computational Chronology of Pi / 683 \\ Appendix III --- Selected Formulae for Pi / 686 \\ Appendix IV --- Translations of Viele and Huygens / 690 \\ Bibliography / 710 \\ Credits / 717 \\ A Pamphlet on Pi / 721 \\ Contents / 723 \\ 1. Pi and Its Friends / 725 \\ 2. Normality of Numbers / 741 \\ 3. Historia Cyclometrica / 753 \\ 4. Demotica Cyclometrica / 771 \\ References / 779 \\ Index / 783", } @Book{Arndt:2011:MC, author = "J{\"o}rg Arndt", booktitle = "Matters Computational", title = "Matters Computational", publisher = pub-SV, address = pub-SV:adr, pages = "xiv + 966", year = "2011", DOI = "https://doi.org/10.1007/978-3-642-14764-7", ISBN = "3-642-14764-X, 3-642-14763-1", ISBN-13 = "978-3-642-14764-7, 978-3-642-14763-0", LCCN = "QA9.58 .A76 2011", bibdate = "Tue Mar 14 15:06:32 MDT 2017", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/agm.bib", abstract = "This book provides algorithms and ideas for computationalists. Subjects treated include low-level algorithms, bit wizardry, combinatorial generation, fast transforms like the Fourier transform, and fast arithmetic for both real numbers and finite fields. Various optimization techniques are described and the actual performance of many given implementations is examined. The focus is on material that does not usually appear in textbooks on algorithms. The implementations are done in C++ and the GP language, written for POSIX-compliant platforms such as the Linux and BSD operating systems.", acknowledgement = ack-nhfb, shorttableofcontents = "Matters Computational \\ Preface \\ Contents \\ Part I: Low level algorithms \\ Part II:Combinatorial generation \\ Part III:Fast transforms \\ Part IV: Fast arithmetic \\ Part V: Algorithms for finite fields \\ Appendix A: The electronic version of the book \\ Appendix B: Machine used for benchmarking \\ Appendix C: The GP language \\ Bibliography \\ Index", subject = "Mathematics", tableofcontents = "Low level algorithms \\ Bit wizardry / 2--101 \\ Permutations and their operations / 102--133 \\ Sorting and searching / 134--152 \\ Data structures / 153--170 \\ Combinatorial generation \\ Conventions and considerations / 172--175 \\ Combinations / 176--193 \\ Compositions / 194--201 \\ Subsets / 202--216 \\ Mixed radix numbers / 217--231 \\ Permutations / 232--276 \\ Permutations with special properties / 277--290 \\ $k$-permutations / 291--294 \\ Multisets / 295--303 \\ Gray codes for string with restrictions / 304--322 \\ Parenthesis strings / 323--338 \\ Integer partitions / 339--353 \\ Set partitions / 354--369 \\ Necklaces and Lyndon words / 370--383 \\ Hadamard and conference matrices / 384--390 \\ Searching paths in directed graphs / 391--408 \\ Fast transforms \\ The Fourier transform / 410--439 \\ Convolution, correlation, and more FFT algorithms / 440--458 \\ The Walsh transform and its relatives / 459--496 \\ The Haar transform / 497--514 \\ The Hartley transform / 515--534 \\ Number theoretic transforms (NTTs) / 535--542 \\ Fast wavelet transforms / 543--548 \\ Fast arithmetic \\ Fast multiplication and exponentiation / 550--566 \\ Root extraction / 567--586 \\ Iterations for the inversion of a function / 587--598 \\ The AGM, elliptic integrals, and algorithms for computing / 599--621 \\ Logarithm and exponential function / 622--640 \\ Computing the elementary functions with limited resources / 641--650 \\ Numerical evaluation of power series / 651--665 \\ Recurrences and Chebyshev polynomials / 666--684 \\ Hypergeometric series / 685--703 \\ Cyclotomic polynomials, product forms, and continued fractions / 704--725 \\ Synthetic Iterations / 726--762 \\ Algorithms for finite fields \\ Modular arithmetic and some number theory / 764--821 \\ Binary polynomials / 822--863 \\ Shift registers / 864--885 \\ Binary finite fields: $ {\rm GF}(2^n) $ / 886--920 \\ The electronic version of the book \\ Machine used for benchmarking \\ The GP language \\ Bibliography \\ Index", } @Book{Bailey:2016:PNG, editor = "David H. Bailey and Jonathan M. Borwein", booktitle = "Pi: the next generation: a sourcebook on the recent history of Pi and its computation", title = "Pi: the next generation: a sourcebook on the recent history of Pi and its computation", publisher = pub-SV, address = pub-SV:adr, pages = "xiv + 507", year = "2016", DOI = "https://doi.org/10.1007/978-3-319-32377-0", ISBN = "3-319-32375-X, 3-319-32377-6 (e-book)", ISBN-13 = "978-3-319-32375-6, 978-3-319-32377-0 (e-book)", LCCN = "QA251", MRclass = "01A75, 11-00, 65-00, 11-06, 65-06", bibdate = "Fri Sep 30 09:43:05 2016", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib", URL = "http://docserver.carma.newcastle.edu.au/1716/; http://lib.myilibrary.com?id=941862", ZMnumber = "1342.01042", acknowledgement = ack-nhfb, subject = "Pi", tableofcontents = "Computation of $\pi$ using arithmetic--geometric mean (1976) / Salamin, Eugene / 1--8 \\ Fast multiple-precision evaluation of elementary functions (1976) / Brent, Richard P. / 9--20 \\ The arithmetic--geometric mean of Gauss (1984) / Cox, David A. / 21--78 \\ The arithmetic--geometric mean and fast computation of elementary functions (1984) / Borwein, J. M. and Borwein, P. B. / 79--96 \\ A simplified version of the fast algorithms of Brent and Salamin (1985) / Newman, D. J. / 97--102 \\ Is pi normal? (1985) / Wagon, S. / 103--107 \\ The computation of $\pi$ to 29,360,000 decimal digits using Borweins' quartically convergent algorithm (1988) / Bailey, David H. / 109--124 \\ Gauss, Landen, Ramanujan, the arithmetic--geometric mean, ellipses, $\pi$, and the Ladies Diary (1988) / Almkvist, Gert (et al.) / 125--150 \\ Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation (1988) / Kanada, Yasumasa / 151--164 \\ Ramanujan and pi (1988) / Borwein, Jonathan M. (et al.) / 165--174 \\ Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi (1989) / Borwein, Jonathan M. (et al.) / 175--195 \\ Pi, Euler numbers, and asymptotic expansions (1989) / Borwein, Jonathan M. (et al.) / 197--205 \\ A spigot algorithm for the digits of $\pi$ (1995) / Rabinowitz, Stanley (et al.) / 207--217 \\ On the rapid computation of various polylogarithmic constants (1997) / Bailey, David H. (et al.) / 219--231 \\ Similarities in irrationality proofs for $\pi$, ln 2, \zeta(2), and \zeta(3) (2001) / Huylebrouck, Dirk / 233--244 \\ Unbounded spigot algorithms for the digits of pi (2006) / Gibbons, Jeremy / 245--257 \\ Mathematics by experiment: Plausible reasoning in the 21st Century (2008) / Bailey, David H. (et al.) / 259--291 \\ Approximations to $\pi$ derived from integrals with nonnegative integrands (2009) / Lucas, Stephen K. / 293--301 \\ Ramanujan's series for 1/$\pi$: A survey (2009) / Baruah, Nayandeep Deka (et al.) / 303--325 \\ The computation of previously inaccessible digits of $\pi$ / Bailey, David H. (et al.) / 327--339 \\ Walking on real numbers (2013) / Artacho, Francisco J. Arag{\'o}n (et al.) / 341--361 \\ Birth, growth and computation of pi to ten trillion digits (2013) / Agarwal, Ravi (et al.) / 363--423 \\ Pi day is upon us again and we still do not know if pi is normal (2014) / Bailey, David H. (et al.) / 425--442 \\ The Life of $\pi$ (2014) / Borwein, Jonathan M. (et al.) / 443--474 \\ I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly (2015) / Borwein, Jonathan M. / 475--499", } @Proceedings{Bailey:2020:AVC, editor = "David H. Bailey and Naomi Simone Borwein and Richard P. Brent and Regina S. Burachik and Judy-anne Heather Osborn and Brailey Sims and Qiji J. Zhu", booktitle = "From Analysis to Visualization: A Celebration of the Life and Legacy of {Jonathan M. Borwein, Callaghan, Australia, September 2017}", title = "From Analysis to Visualization: A Celebration of the Life and Legacy of {Jonathan M. Borwein, Callaghan, Australia, September 2017}", volume = "313", publisher = pub-SV, address = pub-SV:adr, year = "2020", DOI = "https://doi.org/10.1007/978-3-030-36568-4", ISBN = "3-030-36567-0 (print), 3-030-36568-9 (e-book)", ISBN-13 = "978-3-030-36567-7 (print), 978-3-030-36568-4 (e-book)", ISSN = "2194-1009 (print), 2194-1017 (electronic)", LCCN = "????", MRclass = "00B20, 11-06, 26-06, 33-06, 47-06, 49-06, 52-06, 62P05, 91G99, 97-06", bibdate = "Tue Apr 21 10:22:01 MDT 2020", bibsource = "fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/pi.bib", series = "Springer Proceedings in Mathematics \& Statistics", ZMnumber = "07174492", acknowledgement = ack-nhfb, remark = "Book.", subject = "Education / Teaching Methods and Materials / Mathematics; Mathematics / Applied; Mathematics / Mathematical Analysis; Mathematics / Number Theory", subject-dates = "Jonathan Michael Borwein (20 May 1951--2 August 2016)", tableofcontents = "Part I: Applied Analysis, Optimisation, and Convexity \\ Introduction / Regina S. Burachik and Guoyin Li / 3--5 \\ Symmetry and the Monotonicity of Certain Riemann Sums / David Borwein and Jonathan M. Borwein and Brailey Sims / 7--20 \\ Risk and Utility in the Duality Framework of Convex Analysis / R. Tyrrell Rockafellar / 21--42 \\ Characterizations of Robust and Stable Duality for Linearly Perturbed Uncertain Optimization Problems / Nguyen Dinh and Miguel A. Goberna and Marco A. Lopez and Michel Volle / 43--74 \\ Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples / Reinier Diaz Millan and Scott B. Lindstrom and Vera Roshchina / 75--98 \\ Part II: Education \\ Introduction / Naomi Simone Borwein / 101--102 \\ On the Educational Legacies of Jonathan M. Borwein / Naomi Simone Borwein and Judy-anne Heather Osborn / 103--131 \\ How Mathematicians Learned to Stop Worrying and Love the Computer / Keith Devlin / 133--139 \\ Crossing Boundaries: Fostering Collaboration Between Mathematics Educators and Mathematicians in Initial Teacher Education Programmes / Merrilyn Goos / 141--148 \\ Mathematics Education in the Computational Age: Challenges and Opportunities / Kathryn Holmes / 149--152 \\ Mathematics Education for Indigenous Students in Preparation for Engineering and Information Technologies / Collin Phillips and Fu Ken Ly / 153--169 \\ Origami as a Teaching Tool for Indigenous Mathematics Education / Michael Assis and Michael Donovan / 171--188 \\ Dynamic Visual Models: Ancient Ideas and New Technologies / Damir Jungic and Veselin Jungic / 189--201 \\ A Random Walk Through Experimental Mathematics / Eunice Y. S. Chan and Robert M. Corless / 203--226 \\ Part III: Financial Mathematics \\ Introduction / David H. Bailey and Qiji J. Zhu / 229--231 \\ A Holistic Approach to Empirical Analysis: The Insignificance of $P$, Hypothesis Testing and Statistical Significance* / Morris Altman / 233--253 \\ Do Financial Gurus Produce Reliable Forecasts? / David H. Bailey and Jonathan M. Borwein and Amir Salehipour and Marcos Lopez de Prado / 255--274 \\ Entropy Maximization in Finance / Jonathan M. Borwein and Qiji J. Zhu / 275--295 \\ Part IV: Number Theory, Special Functions, and Pi \\ Introduction / Richard P. Brent / 299--302 \\ Binary Constant-Length Substitutions and Mahler Measures of Borwein Polynomials / Michael Baake and Michael Coons and Neil Manibo / 303--322 \\ The Borwein Brothers, Pi and the AGM / Richard P. Brent / 323--347 \\ The Road to Quantum Computational Supremacy / Cristian S. Calude and Elena Calude / 349--367 \\ Nonlinear Identities for Bernoulli and Euler Polynomials / Karl Dilcher / 369--376 \\ Metrical Theory for Small Linear Forms and Applications to Interference Alignment / Mumtaz Hussain and Seyyed Hassan Mahboubi and Abolfazl Seyed Motahari / 377--393 \\ Improved Bounds on Brun's Constant / Dave Platt and Tim Trudgian / 395--406 \\ Extending the PSLQ Algorithm to Algebraic Integer Relations / Matthew P. Skerritt and Paul Vrbik / 407--421 \\ Short Walk Adventures / Armin Straub and Wadim Zudilin / 423--439", }