Two new proofs of the complete monotonicity of a function involving the psi function

TL;DR

This paper provides two novel proofs establishing the condition under which a specific function involving the psi function is completely monotonic on the positive real line.

Contribution

It introduces two new proofs for the necessary and sufficient condition of complete monotonicity of a function involving the psi function.

Findings

The function is completely monotonic if and only if α ≤ 1.

Two distinct proofs are provided for this condition.

The results deepen understanding of the monotonicity properties of functions related to the psi function.

Abstract

In the present paper, we give two new proofs for the necessary and sufficient condition $\alpha\le1$ such that the function $x^\alpha[\ln x-\psi(x)]$ is completely monotonic on $(0,\infty)$.