SHARP INEQUALITIES INVOLVING THE POWER MEAN AND COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND

被引：33
|
作者：
Chu, Y. M. [1 ]
Qiu, S. L. [2 ]
Wang, M. K. [1 ]
机构：
[1] Hunan City Univ, Sch Math & Computat Sci, Yiyang 413000, Peoples R China
[2] Zhejiang Sci Tech Univ, Dept Math, Hangzhou 313018, Zhejiang, Peoples R China
来源：
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS | 2013年 / 43卷 / 05期
关键词：
Complete elliptic integrals; power mean; inequality; HYPERGEOMETRIC-FUNCTIONS; FUNCTIONAL INEQUALITIES; MODULAR EQUATIONS; RAMANUJAN; LANDEN;
D O I：
10.1216/RMJ-2013-43-5-1489
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
In this paper, we prove that M-p(K(r), K(r')) >= K (root 2/2) and M-q(K(r), K(r')) <= K(root 2/2) for all r is an element of (0, 1) if and only if p >= 1 - 4[K(root 2/2)](4)/pi(2) = -3.789... and q <= (log 2)/[log(pi/2) - log K(root 2/2)] = -4.180..., where K(r) = integral(0)(pi/2)(1 - r(2) sin(2)theta) (1/2) d theta is the complete elliptic integral of the first kind, r' = root 1 - r(2), and M-p(x, y) is the power mean of order p of two positive numbers x and y.
引用
收藏
页码：1489 / 1496
页数：8
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