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 $\Gamma(x)$, the $n$-th harmonic number $\sum_{k=1}^n\frac1k$, and the factorial $n!$. 
Feng Qi
math.CA