A completely monotonic function involving the gamma and trigamma functions

Kuwait Journal of Science

Feng Qi¹ 

[1] Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

关键词: Completely monotonic function;    gamma function;    inequality;    logarithmically completely monotonic function;    trigamma function.;   

DOI  :  

学科分类：社会科学、人文和艺术（综合）

来源: Kuwait University * Academic Publication Council

【 摘 要 】

In the paper the author provides necessary and sufficient conditions on $a$ for the function\begin{equation*}\frac{1}{2}\ln(2\pi)-x+\biggl(x-\frac{1}{2}\biggr)\ln x-\ln\Gamma(x)+\frac1{12}{\psi'(x+a)}\end{equation*}and its negative to be completely monotonic on $(0,\infty)$, where $a\ge0$ is a real number, $\Gamma(x)$ is the classical gamma function, and $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ is the digamma function. As applications, some known results and new inequalities are derived.

【 授权许可】

Unknown