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 $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are established. Furthermore, let the sequence $r_n$ be defined by $n!=\sqrt{2πn}(n/e)^n e^{r_n}.$ We establish new estimates for Stirling's formula remainder $r_n.$

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