Complete monotonicity of a function involving the $p$-psi function and alternative proofs

TL;DR

This paper proves that a specific function involving the $p$-psi function is completely monotonic on positive real numbers if and only if a parameter $ extalpha$ is less than or equal to 1, extending previous results.

Contribution

It provides an alternative proof for the complete monotonicity of a function involving the $p$-psi function, generalizing earlier known results.

Findings

The function is completely monotonic if and only if $ extalpha extle 1$.

The proof offers a new perspective on the monotonicity properties of $p$-psi functions.

Generalizes classical monotonicity results to the $p$-analogue setting.

Abstract

In the paper the authors alternatively prove that the function $x^\alpha\big[\ln\frac{px}{x+p+1}-\psi_p(x)\big]$ is completely monotonic on $(0,\infty)$ if and only if $\alpha \le 1$, where $p\in\mathbb{N}$ and $\psi_p(x)$ is the $p$-analogue of the classical psi function $\psi(x)$. This generalizes a known result.