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Commun. Korean Math. Soc. 2010; 25(3): 373-383
Printed September 1, 2010
https://doi.org/10.4134/CKMS.2010.25.3.373
Copyright © The Korean Mathematical Society.
The geometric convexity of a function involving gamma function with applications
Yuming Chu, Xiaoming Zhang, and Zhihua Zhang
Huzhou teachers college, Haining radio and tv University, Zixing municipal school
In this paper, we prove that $\left(\Gamma(x)\right)^{\frac{1}{x-1}}$ is geometrically convex on $(0,\infty).$ As its applications, we obtain some new estimates for $\frac{[\Gamma(x+1)]^{\frac{1}{x}}}{[\Gamma(y+1)]^{\frac{1}{y}}}$.
Keywords: gamma function, geometrically convex function, geometrically concave function, monotonicity
MSC numbers: 26D15
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