A completely monotonic function involving the gamma and trigamma functions

Abstract

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.

Keywords

• Completely monotonic function
• gamma function
• inequality
• logarithmically completely monotonic function
• trigamma function.