Journal of the
Korean Mathematical Society JKMS

In this article, the logarithmically complete monotonicity of some functions such as \begin{align*} &\tfrac1{[\Gamma(x+1)]^{1/x}}, &&\tfrac{[\Gamma(x+1)]^{1/x}}{x^\alpha},& & \tfrac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}& &\text{and} && \tfrac{[{\Gamma(x+\alpha+1)}]^{1/(x+\alpha)}}{[{\Gamma(x+1)}]^{1/x}} \end{align*} for $\alpha\in\mathbb{R}$ on $(-1,\infty)$ or $(0,\infty)$ are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.

Keywords: logarithmically completely monotonic function, completely monotonic function, gamma function, basic property

MSC numbers: Primary 26A48, 33B15; Secondary 26A51, 65R10