Approximations related to the complete <i>p</i>-elliptic integrals-Reference-Cited by-同舟云学术

Approximations related to the complete p-elliptic integrals

Published:2022-01-01 Issue:1 Volume:20 Page:1046-1056
ISSN:2391-5455
Container-title:Open Mathematics
language:en
Short-container-title:

Author:

Zhong Genhong¹,Ma Xiaoyan²,Wang Fei³

Affiliation:

1. Keyi College of Zhejiang Sci-Tech University , Shaoxing 312300 , China

2. Department of Mathematics, Zhejiang Sci-Tech University , Hangzhou 310018 , China

3. Zhejiang Institute of Mechanical and Electrical Engineering , Hangzhou 310053 , China

Abstract

Abstract In this paper, the authors present some monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of

K p

{{\mathscr{K}}}_{p}

E p

{{\mathscr{E}}}_{p}

and the inverse hyperbolic tangent

arth p

{{\rm{arth}}}_{p}

, which is of importance in the computation of the generalized pi and in the elementary proof of Ramanujan’s cubic transformation. By these results, several well-known results for the classical complete elliptic integrals including its bounds and logarithmic inequalities are remarkably improved.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/math-2022-0493/pdf

Reference30 articles.

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4. H. Alzer and S. L. Qiu, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math. 172 (2004), no. 2, 289–312.

5. G. D. Anderson, S. L. Qiu, M. K. Vamanamurthy, and M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math. 192 (2000), no. 1, 1–37.

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