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On the monotonicity and convexity for generalized elliptic integral of the first kind

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we investigate the monotonicity and convexity of the function \( x\mapsto \mathcal {K}_{a}(\sqrt{x})/\log (1+c/\sqrt{1-x})\) on (0, 1) for \((a,c)\in (0,1/2]\times (0,\infty )\), and the log-concavity of the function \(x\mapsto (1-x)^{\lambda }\mathcal {K}_{a}(\sqrt{x})\) on (0, 1) for \(\lambda \in \mathbb {R}\), where \(\mathcal {K}_{a}(r)\) is the generalized elliptic integral of the first kind. These results are the generalization of [1, Theorem 2] and [2, Theorems 1.7 and 1.8], also give an affirmative answer of [3, Problem 3.1].

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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 (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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Chen, Yj., Zhao, Th. On the monotonicity and convexity for generalized elliptic integral of the first kind. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 77 (2022). https://doi.org/10.1007/s13398-022-01211-x

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