Some logarithmically completely monotonic functions related to the gamma function

Details Some logarithmically completely monotonic functions related to the gamma function Some logarithmically completely monotonic functions related to the gamma function

2009-03-30

arxiv.org/abs/0903.5123v1

Feng Qi and Bai-Ni Guo, Some logarithmically completely monotonic functions related to the gamma function, Journal of the Kore

Mathematics

Classical Analysis and ODEs

16 pages

Scientific paper

10.4134/JKMS.2010.47.6.1283

In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[\Gamma(x+1)]^{1/x}}$, $\frac{[{\Gamma(x+\alpha+1)}]^{1/(x+\alpha)}}{[{\Gamma(x+1)}]^{1/x}}$, $\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}$ and $\frac{[\Gamma(x+1)]^{1/x}}{x^\alpha}$ defined in $(-1,\infty)$ or $(0,\infty)$ for given real number $\alpha\in\tr$ 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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