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On the Convexity and Concavity of Generalized Complete Elliptic Integral of the First Kind

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Abstract

In this paper, we study the convexity (concavity) of the function \(x\mapsto {{\,\mathrm{{{{\textsf {\textit{K}}}}}}\,}}_a(\sqrt{x})-\log \left( 1+c/\sqrt{1-x}\right) \) on (0, 1) for \(a\in (0,1/2]\) and \(c\in (0,\infty )\), where \({{\,\mathrm{{{{\textsf {\textit{K}}}}}}\,}}_a(r)\) is the generalized complete elliptic integral of the first kind. This work is an extension of Yang and Tian (Appl Anal Discrete Math 13:240–260, 2019), and also gives a refinement of inequality (Yang and Tian 2019, 0.27) as an application.

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Funding

This work is supported by the National Natural Science Foundation of China (11971142) and the Natural Science Foundation of Zhejiang Province (LY19A010012).

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  1. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Ya-jun Chen & Tie-hong Zhao

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  1. Ya-jun Chen
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Correspondence to Tie-hong Zhao.

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Chen, Yj., Zhao, Th. On the Convexity and Concavity of Generalized Complete Elliptic Integral of the First Kind. Results Math 77, 215 (2022). https://doi.org/10.1007/s00025-022-01755-9

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