Complete monotonicity of a function involving the gamma function and applications

In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $\mathrm{\Gamma }(x)$, the $n$-th harmonic number ${\sum }_{k=1}^n\frac{1}{k}$, and the factorial $n!$.