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A note on an open problem

DOI: 10.5556/j.tkjm.43.2012.281-287

Keywords: Gamma function , $ psi $ function , polygamma function , complete monotonicity

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Abstract:

The function $frac{Gamma(x+1)^{frac1x}}{(x+eta)^{alpha}}$ is logarithmically completely monotonic on $(0,infty)$ for $alphage1$ and $0leetale1$, and is logarithmically completely monotonic in $(-1,0)$ for $01$. This give an answer to an open problem proposed by Feng Qi.

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