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Monotonicity and convexity (concavity) properties for zero-balanced hypergeometric function

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Abstract

In this paper, for a suitable region of (a, b), we establish a necessary and sufficient condition of \(p>0\) such that

$$\begin{aligned} x\mapsto \frac{\log (p/\sqrt{1-x})}{F(a,b;a+b;x)} \end{aligned}$$

is strictly monotonic, convex, or concave on (0, 1), where \(F(a,b;a+b;x)\) represents the zero-balanced hypergeometric function. This extends the recently obtained corresponding results for the cases that \(a=b=1/2\). As applications, several functional inequalities involving zero-balanced hypergeometric function will be obtained.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (11971142) and the Natural Science Foundation of Zhejiang Province (LY24A010011).

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Authors and Affiliations

  1. School of mathematics, Hangzhou Normal University, Hangzhou, 311121, People’s Republic of China

    Tie-Hong Zhao

  2. Department of mathematics, Huzhou University, Huzhou, 313000, People’s Republic of China

    Miao-Kun Wang

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

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Zhao, TH., Wang, MK. Monotonicity and convexity (concavity) properties for zero-balanced hypergeometric function. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 56 (2024). https://doi.org/10.1007/s13398-024-01555-6

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  • DOI: https://doi.org/10.1007/s13398-024-01555-6

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