A completely monotonic function used in an inequality of Alzer

TL;DR

This paper proves that the derivative of a specific function used in an inequality by Alzer is completely monotonic by deriving its integral representation, despite it not being a Stieltjes function.

Contribution

It establishes the complete monotonicity of the derivative of a function related to Alzer's inequality through integral representation techniques.

Findings

Proves $G'$ is completely monotonic.

Derives integral representation of the holomorphic extension of $G$.

Shows $G'$ is not a Stieltjes function.

Abstract

The function $G(x)=(1-\ln x /\ln(1+x))x\ln x$ has been considered by Alzer, Qi and Guo. We prove that $G'$ is completely monotonic by finding an integral representation of the holomorphic extension of $G$ to the cut plane. A main difficulty is caused by the fact that $G'$ is not a Stieltjes function.