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.
引用
收藏
页数：17
相关论文
共 50 条
12345