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Sharp Approximations for Complete p-Elliptic Integral of the Second Kind by Weighted Power Means

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Abstract

In this paper, the well-known double inequality for the complete elliptic integral E(r) of the second kind, which gives sharp approximations of E(r) by power means (or Hölder means), is extended to the complete p-elliptic integral \(E_p(r)\) of the second kind, and thus sharp approximations of \(E_p(r)\) by weighted power means are obtained. This result confirmed the truth of Conjecture I by Barnard, Ricards and Tiedeman in the case when \(a=b=1/p\in (0,1/2)\) and \(c=1\) and also provides a new method to prove the above double inequality of E(r).

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Funding

The author thanks Professor Qiu Songliang for his many valuable suggestions on this manuscript. 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

    Tiehong Zhao

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Correspondence to Tiehong Zhao.

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Communicated by Rosihan M. Ali.

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Zhao, T. Sharp Approximations for Complete p-Elliptic Integral of the Second Kind by Weighted Power Means. Bull. Malays. Math. Sci. Soc. 46, 126 (2023). https://doi.org/10.1007/s40840-023-01523-0

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