SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTION

Title & Authors
SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTION
Qi, Feng; Guo, Bai-Ni;

Abstract
In this article, the logarithmically complete monotonicity of some functions such as $\small{\frac{1}{[\Gamma(x+1)]^{1/x}}$, $\small{\frac{[\Ga}$$\small{mma(x+1)]^{1/x}}{x^\alpha}}$, $\small{\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}}$ and $\small{\frac{[\Gamma(x+\alpha+1)]^{1/(x+\alpha})}{[\Gamma(x+1)^{1/x}}}$ for $\small{\alpha{\in}\mathbb{R}}$ on ($\small{-1,\infty}$) or ($\small{0,\infty}$) 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;
Language
English
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