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Inequalities for zero-balanced hypergeometric functions. (English) Zbl 0826.33003
The authors examine certain monotoneity and convexity properties of the Gaussian zero-balanced hypergeometric functions, and the Euler gamma function. In particular they extend some known properties of the elliptic integral \({\mathcal K} (x)\).

MSC:
33C05 Classical hypergeometric functions, \({}_2F_1\)
33B15 Gamma, beta and polygamma functions
26D07 Inequalities involving other types of functions
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[1] Milton Abramowitz and Irene A. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. · Zbl 0171.38503
[2] Horst Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337 – 346. · Zbl 0802.33001
[3] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Functional inequalities for complete elliptic integrals and their ratios, SIAM J. Math. Anal. 21 (1990), no. 2, 536 – 549. · Zbl 0692.33001 · doi:10.1137/0521029 · doi.org
[4] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Functional inequalities for hypergeometric functions and complete elliptic integrals, SIAM J. Math. Anal. 23 (1992), no. 2, 512 – 524. · Zbl 0764.33009 · doi:10.1137/0523025 · doi.org
[5] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Hypergeometric functions and elliptic integrals, Current topics in analytic function theory, World Sci. Publ., River Edge, NJ, 1992, pp. 48 – 85. · Zbl 0984.33502
[6] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J. Math. 160 (1993), no. 1, 1 – 18. · Zbl 0793.30014
[7] Richard Askey, Ramanujan and hypergeometric and basic hypergeometric series, Ramanujan International Symposium on Analysis (Pune, 1987) Macmillan of India, New Delhi, 1989, pp. 1 – 83. · Zbl 0722.33009
[8] Richard Askey, Handbooks of special functions, A century of mathematics in America, Part III, Hist. Math., vol. 3, Amer. Math. Soc., Providence, RI, 1989, pp. 369 – 391.
[9] L. Baker, \( C\) Mathematical Function Handbook, McGraw-Hill, New York, 1992.
[10] Bruce C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York, 1989. · Zbl 0716.11001
[11] Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. A study in analytic number theory and computational complexity; A Wiley-Interscience Publication. · Zbl 0611.10001
[12] Jonathan M. Borwein and Peter B. Borwein, Inequalities for compound mean iterations with logarithmic asymptotes, J. Math. Anal. Appl. 177 (1993), no. 2, 572 – 582. · Zbl 0783.33001 · doi:10.1006/jmaa.1993.1278 · doi.org
[13] Wolfgang Bühring, Generalized hypergeometric functions at unit argument, Proc. Amer. Math. Soc. 114 (1992), no. 1, 145 – 153. · Zbl 0754.33003
[14] Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Bd LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. · Zbl 0055.11905
[15] Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977.
[16] Ronald J. Evans, Ramanujan’s second notebook: asymptotic expansions for hypergeometric series and related functions, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 537 – 560.
[17] Walter Gautschi, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282 – 292. · Zbl 0239.33003 · doi:10.1137/0505031 · doi.org
[18] D. W. Lozier and F. W. J. Olver, Numerical evaluation of special functions, Mathematics of Computation 1943 – 1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 79 – 125. · Zbl 0815.65030 · doi:10.1090/psapm/048/1314844 · doi.org
[19] Yudell L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. · Zbl 0318.33001
[20] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. · Zbl 0199.38101
[21] S. L. Moshier, Methods and programs for mathematical functions, Ellis Horwood, Chichester, 1989. · Zbl 0701.65011
[22] W. F. Perger, M. Nardin, and A. Bhalla, A numerical evaluator for the generalized hypergeometric function, Comput. Phys. Comm. 77 (1993), 249-254. · Zbl 0854.65015
[23] B. A. Popov and G. S. Tesler, Computation of functions on electronic computers–Handbook, Naukova Dumka, Kiev, 1984. (Russian)
[24] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2, 2nd ed., Gordon & Breach Science Publishers, New York, 1988. Special functions; Translated from the Russian by N. M. Queen. · Zbl 0733.00005
[25] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. · JFM 53.0180.04
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