A note on an open problem

The function $frac{Gamma(x+1)^{frac1x}}{(x+eta)^{alpha}}$ is logarithmically completely monotonic on $(0,infty)$ for $alphage1$ and $0leetale1$, and is logarithmically completely monotonic in $(-1,0)$ for $01$. This give an answer to an open problem proposed by Feng Qi.