A note on inequalities for the ratio of zero-balanced hypergeometric functions
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- by Kendall C. Richards HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 6 (2019), 15-20
Abstract:
Motivated by a question suggested by M. E. H. Ismail in 2017, we present sharp inequalities for the ratio of zero-balanced Gaussian hypergeometric functions. The main theorems generalize known results for complete elliptic integrals of the first kind.References
- Horst Alzer and Kendall Richards, Inequalities for the ratio of complete elliptic integrals, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1661–1670. MR 3601557, DOI 10.1090/proc/13337
- G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1713–1723. MR 1264800, DOI 10.1090/S0002-9947-1995-1264800-3
- G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math. 192 (2000), no. 1, 1–37. MR 1741031, DOI 10.2140/pjm.2000.192.1
- 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. MR 1038906, DOI 10.1137/0521029
- 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. MR 1147875, DOI 10.1137/0523025
- Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR 1462077
- Árpád Baricz, Turán type inequalities for generalized complete elliptic integrals, Math. Z. 256 (2007), no. 4, 895–911. MR 2308896, DOI 10.1007/s00209-007-0111-x
- Roger W. Barnard and Kendall C. Richards, A note on the hypergeometric mean value, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 81–88. MR 1931604, DOI 10.1007/BF03320978
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- Ti-Ren Huang, Song-Liang Qiu, and Xiao-Yan Ma, Monotonicity properties and inequalities for the generalized elliptic integral of the first kind, J. Math. Anal. Appl. 469 (2019), no. 1, 95–116. MR 3857512, DOI 10.1016/j.jmaa.2018.08.061
- M. E. H. Ismail, personal correspondence, May 2016.
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- Zhen-Hang Yang, Wei-Mao Qian, Yu-Ming Chu, and Wen Zhang, On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl. 462 (2018), no. 2, 1714–1726. MR 3774313, DOI 10.1016/j.jmaa.2018.03.005
- Li Yin, Li-Guo Huang, Yong-Li Wang, and Xiu-Li Lin, An inequality for generalized complete elliptic integral, J. Inequal. Appl. , posted on (2017), Paper No. 303, 6. MR 3736599, DOI 10.1186/s13660-017-1578-6
Additional Information
- Kendall C. Richards
- Affiliation: Department of Mathematics and Computer Science, Southwestern University, Georgetown, Texas 78627
- MR Author ID: 311479
- Received by editor(s): January 10, 2019
- Received by editor(s) in revised form: January 12, 2019, January 19, 2019, and February 21, 2019
- Published electronically: May 6, 2019
- Communicated by: Yuan Xu
- © Copyright 2019 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 6 (2019), 15-20
- MSC (2010): Primary 33C05, 33C75; Secondary 26D15
- DOI: https://doi.org/10.1090/bproc/41
- MathSciNet review: 3946862