Some logarithmically completely monotonic functions related to the gamma function

Abstract

In this article, logarithmically complete monotonicity properties of some functions such as $\frac{1}{[\Gamma (x+1){]}^{1/x}}$, $\frac{[\Gamma (x+\alpha +1){]}^{1/(x+\alpha )}}{[\Gamma (x+1){]}^{1/x}}$, $\frac{[\Gamma (x+1){]}^{1/x}}{(x+1{)}^{\alpha }}$ and $\frac{[\Gamma (x+1){]}^{1/x}}{x^{\alpha }}$ 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.

16 pages

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