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

TL;DR

This paper proves that a specific function involving the derivatives of the psi function, related to the gamma function, is completely monotonic over the positive real numbers.

Contribution

It establishes the complete monotonicity of a new function involving tri- and tetra-gamma functions, extending understanding of their properties.

Findings

The function involving $ig[ ext{psi'}(x)ig]^2 + ext{psi''}(x)$ is completely monotonic.

The proof involves properties of the gamma and polygamma functions.

The result contributes to the theory of special functions and their monotonicity properties.

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 $(0,\infty)$.