Complete monotonicity of some functions involving polygamma functions

TL;DR

This paper determines the conditions under which certain functions involving polygamma functions are monotonic and completely monotonic on the positive real axis, expanding understanding of their mathematical properties.

Contribution

It provides necessary and sufficient conditions for the monotonicity and complete monotonicity of functions involving polygamma functions, a novel characterization in this area.

Findings

Established conditions for monotonicity of functions with polygamma functions.

Derived criteria for complete monotonicity on (0,∞).

Enhanced understanding of polygamma function properties.

Abstract

In the present paper, we establish necessary and sufficient conditions for the functions $x^\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert$ and $\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert-x\bigl\lvert\psi^{(i+1)}(x+\beta)\bigr\lvert$ respectively to be monotonic and completely monotonic on $(0,\infty)$, where $i\in\mathbb{N}$, $\alpha>0$ and $\beta\ge0$ are scalars, and $\psi^{(i)}(x)$ are polygamma functions.