A sharp lower bound for the complete elliptic integrals of the first kind

被引：8

作者：

Yang, Zhen-Hang ^{[1
,2
]}

Tian, Jing-Feng ^{[3
]}

Zhu, Ya-Ru ^{[3
]}

机构：

[1] North China Elect Power Univ, Engn Res Ctr Intelligent Comp Complex Energy Syst, Minist Educ, Yonghua St 619, Baoding 071003, Peoples R China

[2] Zhejiang Elect Power Co, Res Inst, Hangzhou 310014, Peoples R China

[3] North China Elect Power Univ, Dept Math & Phys, Yonghua St 619, Baoding 071003, Peoples R China

来源：

REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS | 2020年 / 115卷 / 01期

关键词：

Arithmetic-geometric mean; Logarithmic mean; Complete elliptic integrals of the first kind; Inverse hyperbolic tangent function; NP type power series; Inequality; FUNCTIONAL INEQUALITIES; MONOTONICITY; CONVEXITY;

D O I：

10.1007/s13398-020-00949-6

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Let K(r) be the complete elliptic integrals of the first kind and arthr denote the inverse hyperbolic tangent function. We prove that the inequality 2/pi K(r) > [1 - lambda +lambda (arthr/r)(q)](1/q) holds for r is an element of (0, 1) 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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页数：17

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