Mean Value Inequalities for the Digamma Function

被引：0

作者：

Alzer, H. ^{[1
]}

Kwong, M. K. ^{[2
]}

机构：

[1] Morsbacher Str 10, D-51545 Waldbrol, Germany

[2] Hong Kong Polytech Univ, Dept Appl Math, Hong Kong, Peoples R China

来源：

ANALYSIS MATHEMATICA | 2023年 / 49卷 / 01期

关键词：

digamma function; mean value; functional inequality; Euler's constant;

D O I：

10.1007/s10476-023-0206-6

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Let psi be the digamma function and let L(a, b) = (b - a)/ log(b/a) be the logarithmic mean of a and b. We prove that the inequality(*) (b - a)psi( root ab) < (L(a, b) - a)psi(a) + (b - L(a, b))psi (b)holds for all real numbers a and b with b > a > alpha(0). Here, alpha(0) asymptotic to 0.56155 is the only positive solution of50 psi'(x) + 3x psi''(x) = 0.The constant lower bound alpha 0 is best possible. This refines a result of Chu, Zhang and Tang, who showed that (*) is valid for b > a >= 2. Moreover, we prove that the following companion to (*) holds for all a and b with b > a > 0,(L(a,b)-a) psi (a)+(b-L(a,b))psi(b)< (b-a)psi (a+b/2)

引用

页码：1 / 17

页数：17

共 7 条

[1] On some inequalities for the gamma and psi functions [J].