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A sharp lower bound for the complete elliptic integrals 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

Let \({\mathcal {K}}\left( r\right) \) be the complete elliptic integrals of the first kind and \(\text{ arth }r\) denote the inverse hyperbolic tangent function. We prove that the inequality

$$\begin{aligned} \frac{2}{\pi }{\mathcal {K}}\left( r\right) >\left[ 1-\lambda +\lambda \left( \frac{\text{ arth }r}{r}\right) ^{q}\right] ^{1/q} \end{aligned}$$

holds for \(r\in \left( 0,1\right) \) with the best constants \(\lambda =3/4\) and \(q=1/10\). This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic–geometric mean in terms of logarithmic and arithmetic means.

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Authors and Affiliations

  1. Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, Baoding, 071003, People’s Republic of China

    Zhen-Hang Yang

  2. Zhejiang Electric Power Company Research Institute, Hangzhou, 310014, Zhejiang, People’s Republic of China

    Zhen-Hang Yang

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

    Jing-Feng Tian & Ya-Ru Zhu

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  1. Zhen-Hang Yang
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  2. Jing-Feng Tian
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Correspondence to Jing-Feng Tian.

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Dedicated to the 60th anniversary of Zhejiang Electric Power Company Research Institute.

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This work was supported by the Fundamental Research Funds for the Central Universities under Grant 2015ZD29, Grant 13ZD19, and Grant MS117.

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Yang, ZH., Tian, JF. & Zhu, YR. A sharp lower bound for the complete elliptic integrals of the first kind. RACSAM 115, 8 (2021). https://doi.org/10.1007/s13398-020-00949-6

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