A completely monotonic function involving the tri- and tetra-gamma functions

Abstract

The psi function $\psi(x)$ is defined by $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ and $\psi^{(i)}(x)$ for $i\in\mathbb{N}$ denote the polygamma functions, where $\Gamma(x)$ is the gamma function. In this paper we prove that a function involving the difference between $[\psi'(x)]^2+\psi''(x)$ and a proper fraction of $x$ is completely monotonic on $(0,\infty)$.