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Sub- and superadditive properties of Euler's gamma function

Author(s): Horst Alzer
Journal: Proc. Amer. Math. Soc. 135 (2007), 3641-3648.
MSC (2000): Primary 33B15, 39B62; Secondary 26D15
Posted: August 6, 2007
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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)$.

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Additional Information:

Horst Alzer
Affiliation: Morsbacher Str. 10, D-51545 Waldbröl, Germany
Email: alzerhorst@freenet.de

DOI: 10.1090/S0002-9939-07-09057-0
PII: S 0002-9939(07)09057-0
Keywords: Gamma and psi functions, sub- and superadditive, convex, inequalities.
Received by editor(s): September 5, 2006
Posted: August 6, 2007
Communicated by: Andreas Seeger
Copyright of article: Copyright 2007, American Mathematical Society

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