  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.
