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Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind

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Abstract

Let \(\mathcal {K}\left( r\right) \) be the complete elliptic integral of the first kind defined on \(\left( 0,1\right) \). By virtue of the auxiliary function \(H_{f,g}=\left( f^{\prime }/g^{\prime }\right) g-f\), we prove that the function

$$\begin{aligned} Q_{p}\left( x\right) =\frac{\ln \left( p/\sqrt{1-x}\right) }{\mathcal {K} \left( \sqrt{x}\right) } \end{aligned}$$

is strictly convex on \(\left( 0,1\right) \) if and only if \(0<p\le 4\), thus answering a conjecture. Moreover, we completely described the monotonicity of \(Q_{p}\left( x\right) \) on \(\left( 0,1\right) \) for different \(p\in \left( 0,\infty \right) \).

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Authors and Affiliations

  1. Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding, 071003, People’s Republic of China

    Jing-Feng Tian

  2. Department of Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, 310014, People’s Republic of China

    Zhen-Hang Yang

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Tian, JF., Yang, ZH. Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind. Results Math 78, 29 (2023). https://doi.org/10.1007/s00025-022-01799-x

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