Monotonicity, convexity and inequalities involving zero-balanced Gaussian hypergeometric function

Li Xu; Lu Chen; Ti-Ren Huang; Li Xu; Lu Chen; Ti-Ren Huang

doi:10.3934/math.2022692

AIMS Mathematics

2022, Volume 7, Issue 7: 12471-12482. doi: 10.3934/math.2022692

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Monotonicity, convexity and inequalities involving zero-balanced Gaussian hypergeometric function

Li Xu ¹,
Lu Chen ²,
Ti-Ren Huang ^{3
,
,}

1.
Jinhua Radio and Television University, Jinhua 321000, China
2.
Wuxi Vocational Institute of Commerce, Wuxi 214153, China
3.
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received: 04 January 2022 Revised: 16 April 2022 Accepted: 21 April 2022 Published: 26 April 2022
MSC : 33C05, 26D20

We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the zero-balanced Gaussian hypergeometric function $ F(a, b; a+b; x) $.
- Gaussian hypergeometric function,
- monotonicity,
- convexity and concavity
Citation: Li Xu, Lu Chen, Ti-Ren Huang. Monotonicity, convexity and inequalities involving zero-balanced Gaussian hypergeometric function[J]. AIMS Mathematics, 2022, 7(7): 12471-12482. doi: 10.3934/math.2022692

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Abstract

We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the zero-balanced Gaussian hypergeometric function $ F(a, b; a+b; x) $.

References

[1]	G. E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and ite Applications, Cambridge University Press, 1999.
[2]	H. Alzer, K. Richards, A note on a function involving complete elliptic integrals: Monotonicity, convexity, inequalities, Anal. Math., 41 (2015), 133–139. http://dx.doi.org/10.1007/s10476-015-0201-7 doi: 10.1007/s10476-015-0201-7
[3]	M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, New York: Dover Publications, Inc., 1965.
[4]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal invariants, inequalities, and quasiconformal mappings, New York: John Wiley & Sons, 1997.
[5]	G. D. Anderson, S. L. Qiu, M. K. Vamanamurthy, Elliptic integral inequalities, with applications, Constr. Approx., 14 (1998), 195–207. http://dx.doi.org/10.1007/s003659900070 doi: 10.1007/s003659900070
[6]	G. D. Anderson, S. L. Qiu, M. K. Vamanamurthy, M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math., 192 (2000), 1–37. http://dx.doi.org/10.2140/pjm.2000.192.1 doi: 10.2140/pjm.2000.192.1
[7]	R. Balasubramanian, S. Ponnusamy, M. Vuorinen, Functional inequalities for quotients of hypergeometric functions, J. Math. Anal. Appl., 218 (1998), 256–268. https://doi.org/10.1006/jmaa.1997.5776 doi: 10.1006/jmaa.1997.5776
[8]	Y. C. Han, C. Y. Cai, T. R. Huang, Monotonicity, convexity properties and inequalities involving Gaussian hypergeometric functions with applications, AIMS Math., 7 (2022), 4974–4991. http://dx.doi.org/10.3934/math.2022277 doi: 10.3934/math.2022277
[9]	W. A. Day, On monotonicity of the relaxation functions of viscoelastic materials, Proc. Cambridge Philos. Soc., 67 (1970), 503–508. https://doi.org/10.1017/S0305004100045771 doi: 10.1017/S0305004100045771
[10]	J. F. C. Kingman, An introduction to probability theory and its applications, J. R. Stat. Soc. Ser. A, 135 (1972), 430. https://doi.org/10.2307/2344620 doi: 10.2307/2344620
[11]	T. R. Huang, S. L. Qiu, X. Y. Ma, Monotonicity properties and Inequalities for the generalized elliptic integral of the first of kind, J. Math. Anal. Appl., 469 (2019), 95–116. http://dx.doi.org/10.1016/j.jmaa.2018.08.061 doi: 10.1016/j.jmaa.2018.08.061
[12]	T. R. Huang, S. Y. Tan, X. H.Zhang, Monotonicity, convexity, and inequalities for the generalized elliptic integrals, J. Inequal. Appl., 2017 (2017), 278. http://dx.doi.org/10.1186/s13660-017-1556-z doi: 10.1186/s13660-017-1556-z
[13]	T. R. Huang, L. Chen, S. Y. Tan, Y. M. Chu, Monotonicity, Convexity and Bounds Involving the Beta and Ramanujan $R$-function, J. Math. Inequal., 15 (2021), 615–628. http://dx.doi.org/10.7153/jmi-2021-15-45 doi: 10.7153/jmi-2021-15-45
[14]	T. R. Huang, S. Y. Tan, X. Y. Ma, Y. M. Chu, Monotonicity properties and bounds for the complete $p$-elliptic integrals, J. Inequal. Appl., 2018 (2018), 239. http://dx.doi.org/10.1186/s13660-018-1828-2 doi: 10.1186/s13660-018-1828-2
[15]	S. Ponnusamy, M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric function, Mathematika, 44 (1997), 278–301. http://dx.doi.org/10.1112/S0025579300012602 doi: 10.1112/S0025579300012602
[16]	S. L. Qiu, X. Y. Ma, Y. M. Chu, Sharp Landen transformation inequalities for hypergeometric functions, with applications, J. Math. Anal. Appl., 474 (2019), 1306–1337. https://doi.org/10.1016/j.jmaa.2019.02.018 doi: 10.1016/j.jmaa.2019.02.018
[17]	S. Simi$\acute{c}$, M. Vuorinen, Landen inequalities for zero-balanced hypergeometric functions, Abstr. Appl. Anal., 2012 (2012), 932061. https://doi.org/10.1155/2012/932061 doi: 10.1155/2012/932061
[18]	Y. Q. Song, P. G. Zhou, Y. M. Chu, Inequalities for the Gaussian hypergeometric function, Sci. China Math., 57 (2014), 2369–2380. https://doi.org/10.1007/s11425-014-4858-3 doi: 10.1007/s11425-014-4858-3
[19]	Y. L. Luke, Book review, In: J. Wimp, Sequence transformations and their applications, SIAM Review, 24 (1982), 489–490. https://doi.org/10.1137/1024115
[20]	M. K. Wang, Y. M. Chu, Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl., 21 (2018), 521–537. https://doi.org/10.7153/mia-2018-21-38 doi: 10.7153/mia-2018-21-38
[21]	M. K. Wang, Y. M. Chu, Y. P. Jiang, Ramanujan's cubic transformation inequalities for zero-balanced hypergeometric functions, Rocky Mountain J. Math., 46 (2016), 679–691. https://doi.org/10.1216/RMJ-2016-46-2-679 doi: 10.1216/RMJ-2016-46-2-679
[22]	M. K. Wang, W. Zhang, Y. M. Chu, Monotonicity, convexity and inequalities involving the generalized elliptic integrals, Acta. Math. Sci., 39 (2019), 1440–1450. https://doi.org/10.1007/s10473-019-0520-z doi: 10.1007/s10473-019-0520-z
[23]	T. H. Zhao, M. K. Wang, W. Zhang, Y. M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018 (2018), 251. https://doi.org/10.1186/s13660-018-1848-y doi: 10.1186/s13660-018-1848-y
[24]	T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Math., 5 (2020), 4512–4528. https://doi.org/10.3934/math.2020290 doi: 10.3934/math.2020290
[25]	T. H. Zhao, Z. Y. He, Y. M. Chu, On some refinements for inequalities involving zero-balanced hypergeometric function, AIMS Math., 5 (2020), 6479–6495. https://doi.org/10.3934/math.2020418 doi: 10.3934/math.2020418
[26]	S. S. Zhou, G. Farid, C. Y. Jung, Convexity with respect to strictly monotone function and Riemann-Liouville fractional Fejér-Hadamard inequalities, AIMS Math., 6 (2021), 6975–6985. https://doi.org/10.3934/math.2021409 doi: 10.3934/math.2021409
[27]	X. H. Zhang, Monotonicity and functional inequalities for the complete $p$-elliptic integrals, J. Math. Anal. Appl., 453 (2017), 942–953. https://doi.org/10.1016/j.jmaa.2017.04.025 doi: 10.1016/j.jmaa.2017.04.025

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