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In this paper, the logarithmically complete monotonicity property for afunctions involving $q$-gamma function is investigated for $q\in(0,1).$ Asapplications of this results, some new inequalities for the $q$-gamma functionare established. Furthermore, let the sequence $r_n$ be defined by$$n!=\sqrt{2\pi n}(n/e)^n e^{r_n}.$$ We establish new estimates for Stirling'sformula remainder $r_n.$

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