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Sub- and superadditive properties of Euler's gamma function
Authors:Horst Alzer
Institution:Morsbacher Str. 10, D-51545 Waldbröl, Germany
Abstract:Let $ \alpha>0$ and $ 0<c \neq 1$ be real numbers. The inequality

$\displaystyle \Bigl(\frac{\Gamma(x+y+c)}{\Gamma(x+y)}\Bigr)^{1/\alpha}< \Bigl(\... ...mma(x)}\Bigr)^{1/\alpha}+ \Bigl(\frac{\Gamma(y+c)}{\Gamma(y)}\Bigr)^{1/\alpha} $

holds for all positive real numbers $ x, y$ if and only if $ \alpha\geq \max(1,c)$. The reverse inequality is valid for all $ x,y>0$ if and only if $ \alpha\leq \min(1,c)$.

Keywords:Gamma and psi functions  sub- and superadditive  convex  inequalities  
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