A note on an open problem

The function $\frac{\Gamma(x+1)^{\frac1x}}{(x+\beta)^{\alpha}}$ is logarithmically completely monotonic on $(0,\infty)$ for $\alpha\ge1$ and $0\le\beta\le1$, and is logarithmically completely monotonic in $(-1,0)$ for $0<\alpha\le \frac{2\beta}{1+2\beta}$ and $\beta>1$. This give an answer to an ope... View More