Some logarithmically completely monotonic functions related to the gamma function

In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[Γ(x+1)]^{1/x}}$, $\frac{[{Γ(x+α+1)}]^{1/(x+α)}}{[{Γ(x+1)}]^{1/x}}$, $\frac{[Γ(x+1)]^{1/x}}{(x+1)^α}$ and $\frac{[Γ(x+1)]^{1/x}}{x^α}$ defined in $(-1,\infty)$ or $(0,\infty)$ for given real number $α\in\tr$ are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.

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