A Class of logarithmically completely monotonic functions relating the $q$-gamma function and applications

In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in (0,1)\mathrm{.}$ As applications of this results, some new inequalities for the $q$-gamma function are established. Furthermore, let the sequence $r_n$ be defined by $$n!=\sqrt{2\pi n}(n\mathrm{/}e{)}^n{e}^{r_n}\mathrm{.}$$ We establish new estimates for Stirling’s formula remainder $r_n\mathrm{.}$