Complete monotonicity properties of a function involving the polygamma function

TL;DR

This paper investigates the complete monotonicity of a function involving the polygamma function, deriving inequalities and properties for different orders of derivatives, enhancing understanding of special functions in mathematical analysis.

Contribution

It introduces new monotonicity results for a class of functions involving the polygamma function, distinguishing cases based on even and odd derivatives.

Findings

Established complete monotonicity for specific cases of the function.

Derived new inequalities involving polygamma functions.

Enhanced understanding of monotonicity properties of special functions.

Abstract

In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: n\in \mathbb{N}_0 \}$ and $k\in \{ 2n+1: n\in \mathbb{N}_0 \}$. Subsequently, we deduce some inequalities involving the polygamma functions.