A completely monotonic function used in an inequality of Alzer

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 $G^{\prime }$ 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 $G^{\prime }$ is not a Stieltjes function.

13 pages