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Monotonicity and concavity properties of the Gaussian hypergeometric functions, with applications

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Abstract

This paper deals with the monotonicity and concavity properties of certain functions involving the Gaussian hypergeometric function. With these results, we not only obtain sharp bounds for the ratio of hypergeometric functions which extend recently discovered inequalities for k-balanced hypergeometric functions, and but also give an affirmative answer to an open problem proposed by Qiu and Vuorinen. In addition, as by-products, some monotonicity theorems for complete p-elliptic integrals and inequalities for generalized Grötzsch ring function are established.

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References

  1. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, 1964.

    MATH  Google Scholar 

  2. H. Alzer. Sharp inequalities for the complete elliptic integral of the first kind. Math. Proc. Cambridge Philos. Soc., 124(1998), no.2, 30–314.

    MathSciNet  MATH  Google Scholar 

  3. H. Alzer, S.-L. Qiu. Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math., 172(2004), no.2, 289–312.

    MathSciNet  MATH  Google Scholar 

  4. H. Alzer, K. C. Richards. Inequalities for the ratio of complete elliptic integrals. Proc. Amer. Math. Soc., 145(2017), no.4, 1661–1670.

    MathSciNet  MATH  Google Scholar 

  5. G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, M. Vuorinen. Inequalities for zero-balanced hypergeometric functions. Trans. Amer. Math. Soc., 347(1995), no.5, 1713–1723.

    MathSciNet  MATH  Google Scholar 

  6. G. D. Anderson, P. Duren, M. K. Vamanamurthy. An inequality for complete elliptic integrals. J. Math. Anal. Appl., 182(1994), no.1, 257-259.

    MathSciNet  MATH  Google Scholar 

  7. G.D. Anderson, S.-L. Qiu, M.K. Vamanamurthy. Elliptic integral inequalities, with applications. Constr. Approx., 14(1998), no.2, 195–207.

    MathSciNet  MATH  Google Scholar 

  8. G.D. Anderson, S.-L. Qiu, M.K. Vamanamurthy, M. Vuorinen. Generalized elliptic integrals and modular equations. Pacific J. Math., 192(2000), no.1, 1–37.

    MathSciNet  Google Scholar 

  9. G.D. Anderson, S.-L. Qiu, M. Vuorinen. Modular equations and distortion functions. Ramanujan J., 18(2009), no.2, 147–169.

    MathSciNet  MATH  Google Scholar 

  10. G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal., 21(1990), no.2, 536-549.

    MathSciNet  MATH  Google Scholar 

  11. G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Conformal Invariants, Inequalities, and Quasiconformal Maps, Wiley & Sons, New York, 1997.

    MATH  Google Scholar 

  12. G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Topics in special functions II. Conform. Geom. Dyn., 11(2007), 250–270.

    MathSciNet  MATH  Google Scholar 

  13. G. Andrews, R. Askey, R. Roy. Special functions. Encyclopedia of Mathematics and Its Applications. Vol. 70, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  14. R. W. Barnard, K. C. Richards, E. N. Sliheet. On sharp bounds for ratios of \(k\)-balanced hypergeometric functions. Proc. Amer. Math. Soc., 148(2020), no.2, 777–786.

    MathSciNet  MATH  Google Scholar 

  15. Á. Baricz. Turán type inequalities for hypergeometric functions. Proc. Amer. Math. Soc., 136(2008), no.9, 3223–3229.

    MathSciNet  MATH  Google Scholar 

  16. R. Bhatia, R.-C. Li. An interpolating family of means. Commun. Stoch. Anal., 6(2012), no.1, 15–31.

    MathSciNet  MATH  Google Scholar 

  17. B.A. Bhayo, M. Vuorinen. On generalized complete elliptic integrals and modular functions. Pro. Edinb. Math. Soc.(2), 55(2012), no.3, 591–611.

    MathSciNet  MATH  Google Scholar 

  18. J.M. Borwein and P.B. Borwein. Pi and the AGM. John Wiley & Sons, New York, 1987.

    MATH  Google Scholar 

  19. P. Duren, J. Pfaltzgraff. Robin capacity and Extremal length. J. Math. Anal. Appl., 179(1993), no.1, 110-119.

    MathSciNet  MATH  Google Scholar 

  20. A. Dyachenko, D. Karp. Ratios of the Gauss hypergeometric functions with parameters shifted by integers: more on integral representations. Lobachevskii J. Math., 42(2021), no. 12, 2764-2776.

    MathSciNet  MATH  Google Scholar 

  21. P. Hasto, S. Ponnusamy, M. Vuorinen. Starlikeness of the Gaussian hypergeometric functions. Complex Var. Elliptic Equ., 55(2010), no. 1-3, 173–184.

    MathSciNet  MATH  Google Scholar 

  22. R. Küstner. Mapping properties of hypergeometric functions and convolutions of Starlike or convex functions of order \(\alpha \). Comput. Methods Funct. Theory, 2(2002), no.2, 597–610.

    MathSciNet  MATH  Google Scholar 

  23. V. Heikkala, H. Lindén, M.K. Vamanamurthy, M. Vuorinen. Generalized elliptic integrals and the Legendre \({\cal{M}}\)-function. J. Math. Anal. Appl., 338(2008), no.1, 223–243.

    MathSciNet  MATH  Google Scholar 

  24. V. Heikkala, M. K. Vamanamurthy, M. Vuorinen. Generalized elliptic integrals. Comput. Methods Funct. Theory, 9(2009), no.1, 75–109.

    MathSciNet  MATH  Google Scholar 

  25. T.-R. Huang, S.-L. Qiu, X.-Y. Ma. Monotonicity properties and inequalities for the generalized elliptic integral of the first kind. J. Math. Anal. Appl., 469(2019), no.1, 95–116.

    MathSciNet  MATH  Google Scholar 

  26. R. Nishimura (2019). Monotonicity of asymptotic relations for generalized hypergeometric functions. J. Math. Anal. Appl., 480, no.1, 123377, 16 pp.

    MathSciNet  MATH  Google Scholar 

  27. F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (Eds.). NIST Handbook of Mathematical Functions. Cambridge Univ.Press, Cambridge, 2010.

  28. S. Ponnusamy and M. Vuorinen. Asymptotic expansions and inequalities for hypergeometric functions. Mathematika, 44(1997), no.2, 278-301.

    MathSciNet  MATH  Google Scholar 

  29. S. Ponnusamy, M. Vuorinen. Univalence and convexity properties for Gaussian hypergeometric functions. Rocky Mountain J. Math., 31(2001), no.1, 327-353.

    MathSciNet  MATH  Google Scholar 

  30. S.-L. Qiu, X.-Y. Ma, Y.-M. Chu. Sharp Landen transformation inequalities for hypergeometric functions, with applications. J. Math. Anal. Appl., 474(2019), no.2, 1306–1337.

    MathSciNet  MATH  Google Scholar 

  31. S.-L. Qiu, X.-Y. Ma, T.-R. Huang. Sharp approximations for the Ramanujan constant. Constr. Approx., 51(2020), no.2, 303–330.

    MathSciNet  MATH  Google Scholar 

  32. S.-L. Qiu, L-Y. Ren. Sharp estimates for Hübner’s upper bound function with applications. Appl. Math. J. Chinese Univ. Ser. B, 25(2010), no.2, 227–235.

    MathSciNet  MATH  Google Scholar 

  33. S.L. Qiu, M.K. Vamanamurthy. Elliptic integrals and the modulus of Grötzsch ring. PanAmer. Math. J. 5(1995), no.2, 41–60.

    MathSciNet  MATH  Google Scholar 

  34. S.-L. Qiu, M. K. Vamanamurthy and M. Vuorinen. Some inequalities for the Hersch-Pfluger distortion function. J. Inequal. Appl., 4(1999), no.2, 115–139.

    MathSciNet  MATH  Google Scholar 

  35. S.-L. Qiu, M. Vuorinen. Landen inequalities for hypergeometric functions. Nagoya Math. J., 154(1999), 31–56.

    MathSciNet  MATH  Google Scholar 

  36. S.-L. Qiu, M. Vuorinen. Infinite products and the normalized quotients of hypergeometric functions. SIAM J. Math. Anal., 30(1999), no.5, 1057–1075.

    MathSciNet  MATH  Google Scholar 

  37. S.-L. Qiu, M. Vuorinen. Chapter 14 Special functions in geometric function theory. In: Handbook of Complex Analysis: Geometric Function Theory, Vol.2, Elsevier Sci. B. V., Amsterdam, 2005, 621–659.

  38. E. D. Rainville. Special functions. Chelsea Publishing Company, New York, 1960.

    MATH  Google Scholar 

  39. K. C. Richards. A note on inequalities for the ratio of zero-balanced hypergeometric functions. Proc. Amer. Math. Soc. Ser. B, 6(2019), 15–20.

    MathSciNet  MATH  Google Scholar 

  40. L.C. Shen. A note on Ramanujan’s identities involving the hypergeometric function \(_{2}F_{1}(1/6,5/6;1;z)\). Ramanujan J., 30(2013), no.2, 211–222.

    MathSciNet  Google Scholar 

  41. S. Takeuchi. A new form of the generalized complete elliptic integrals. Kodai Math. J., 39(2016), no.1, 202–226.

    MathSciNet  MATH  Google Scholar 

  42. S. Takeuchi. Multiple-angle formulas of generalized trigonometric functions with two parameters. J. Math. Anal. Appl., 444(2016), no.2, 1000–1014.

    MathSciNet  MATH  Google Scholar 

  43. S. Takeuchi. Legendre-type relations for generalized complete elliptic integrals. J. Class. Anal., 9(2016), no.1, 35–42.

    MathSciNet  MATH  Google Scholar 

  44. S. Takeuchi. Complete \(p\)-elliptic integrals and a computation formula of \(\pi _{p}\) for \(p=4\). Ramanujan J., 46(2018), no.2, 309–321.

    MathSciNet  MATH  Google Scholar 

  45. M. Vuorinen. Singular values, Ramanujan modular equations, and Landen transformations. Studia Math., 121(1996), no.3, 221–230.

    MathSciNet  MATH  Google Scholar 

  46. M.-K. Wang, Y.-M. Chu. Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl., 402(2013), no.1, 119–126.

    MathSciNet  MATH  Google Scholar 

  47. M.-K. Wang, Y.-M. Chu, Y.-Q. Song. Ramanujan’s cubic transformation and generalized modular equation. Sci. China. Math., 58(2015), no.11, 2387–2404.

    MathSciNet  MATH  Google Scholar 

  48. M.-K. Wang, Y.-M. Chu, Y.-Q. Song. Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput., 276(2016), 44-60.

    MathSciNet  MATH  Google Scholar 

  49. M.-K. Wang, Y.-M. Chu, W. Zhang. Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J., 49(2019), no.3, 653–668.

    MathSciNet  MATH  Google Scholar 

  50. M.-K. Wang, Y.-M. Chu, W. Zhang. Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl., 22(2019), no.2, 601–617.

    MathSciNet  MATH  Google Scholar 

  51. M.-K. Wang, Y.-M. Li, Y.-M. Chu. Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J., 46(2018), no.1, 189–200

    MathSciNet  MATH  Google Scholar 

  52. M.-K. Wang, S.-L. Qiu, Y.-M. Chu. Infinite series formula for Hübner upper bound function with application to Hersch-Pfluger distortion function. Math. Inequal. Appl., 21(2018), no.3, 629–648.

    MathSciNet  MATH  Google Scholar 

  53. G.-D. Wang, X.-H. Zhang, Y.-M. Chu. Inequalities for the generalized elliptic integrals and modular functions. J. Math. Anal. Appl., 331(2007), no.2, 1275–1283.

    MathSciNet  MATH  Google Scholar 

  54. Zh.-H. Yang. Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means. J. Math. Anal. Appl., 467(2018), no.1, 446–461.

    MathSciNet  MATH  Google Scholar 

  55. Zh.-H. Yang, Y.-M. Chu, M.-K. Wang. Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl., 428(2015), no.1, 587–604.

    MathSciNet  MATH  Google Scholar 

  56. Zh.-H. Yang, J.-F. Tian. Convexity and monotonicity for elliptic integrals of the first kind and applications. Appl. Anal. Discrete. Math., 13(2019), no.1, 240–260.

    MathSciNet  MATH  Google Scholar 

  57. Zh.-H. Yang, J.-F. Tian. Sharp inequalities for the generalized elliptic integrals of the first kind. Ramanujan J., 48(2019), no.1, 91–116.

    MathSciNet  MATH  Google Scholar 

  58. Zh.-H. Yang, J.-F. Tian, M.-K. Wang. A positive answer to Bhatia-Li conjecture on the monotonicity for a new mean in its parameter. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114(2020), no.3, Paper No. 126, 22 pp.

  59. L. Yin, X.-L. Lin, F. Qi. Monotonicity, convexity and inequalities related to complete \((p,q,r)\)-elliptic integrals and generalized trigonometric functions. Publ. Math. Debrecen, 97(2020), no.1-2, 181–199.

    MathSciNet  MATH  Google Scholar 

  60. M. Yoshida. Fuchsian differential equations. with special emphasis on the Gauss-Schwarz theory. Aspects of Mathematics, E11. Friedr. Vieweg & Sohn, Braunschweig, 1987.

  61. X.-H. Zhang. Monotonicity and functional inequalities for the complete \(p\)-elliptic integrals. J. Math. Anal. Appl., 453(2017), no.2, 942–953.

    MathSciNet  MATH  Google Scholar 

  62. X.-H. Zhang, G.-D. Wang, Y.-M. Chu. Remarks on generalized elliptic integrals. Proc. Roy. Soc. Edinburgh Sect. A, 139(2009), no.2, 417–426.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Huzhou University, Huzhou, 313000, People’s Republic of China

    Miao-Kun Wang & Yu-Ming Chu

  2. Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, People’s Republic of China

    Tie-Hong Zhao

  3. School of Science, Changzhou Institute of Technology, Changzhou, 213032, People’s Republic of China

    Xue-Jing Ren

  4. Institute for Advanced Study Honoring Chen Jian Gong, Hangzhou Normal University, Hangzhou, 311121, People’s Republic of China

    Yu-Ming Chu

  5. School of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China

    Zai-Yin He

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  1. Miao-Kun Wang
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  2. Tie-Hong Zhao
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Correspondence to Miao-Kun Wang.

Additional information

Communicated by Rahul Roy.

The research was supported by the Natural Science Foundation of China (Grant Nos. 11701176, 11901061, 11971142) and the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010012)

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Wang, MK., Zhao, TH., Ren, XJ. et al. Monotonicity and concavity properties of the Gaussian hypergeometric functions, with applications. Indian J Pure Appl Math 54, 1105–1124 (2023). https://doi.org/10.1007/s13226-022-00325-7

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  • DOI: https://doi.org/10.1007/s13226-022-00325-7

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