Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind

被引：6

作者：

Tian, Jing-Feng ^{[1
]}

Yang, Zhen-Hang ^{[2
]}

机构：

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

[2] State Grid Zhejiang Elect Power Co, Res Inst, Dept Sci & Technol, Hangzhou 310014, Peoples R China

来源：

RESULTS IN MATHEMATICS | 2023年 / 78卷 / 01期

关键词：

Complete elliptic integral of the first kind; hypergeometric function; convexity; inequality; HYPERGEOMETRIC-FUNCTIONS; FUNCTIONAL INEQUALITIES;

D O I：

10.1007/s00025-022-01799-x

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let K(r) be the complete elliptic integral of the first kind defined on (0, 1). By virtue of the auxiliary function H-f,H-g = (f'/g') g - f, we prove that the function Q(p) (x) = ln (p root 1 - x)/K(root x) is strictly convex on (0, 1) if and only if 0 < p <= 4, thus answering a conjecture. Moreover, we completely described the monotonicity of Q(p) (x) on (0, 1) for different p is an element of (0, infinity).

引用

页数：18

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