Sub- and superadditive properties of Euler’s gamma function
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Abstract:
Let $\alpha >0$ and $0<c \neq 1$ be real numbers. The inequality \[ \Bigl (\frac {\Gamma (x+y+c)}{\Gamma (x+y)}\Bigr )^{1/\alpha }< \Bigl (\frac {\Gamma (x+c)}{\Gamma (x)}\Bigr )^{1/\alpha }+ \Bigl (\frac {\Gamma (y+c)}{\Gamma (y)}\Bigr )^{1/\alpha } \] 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 $\alpha \leq \min (1,c)$.References
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Additional Information
- Horst Alzer
- Affiliation: Morsbacher Str. 10, D-51545 Waldbröl, Germany
- MR Author ID: 238846
- Email: alzerhorst@freenet.de
- Received by editor(s): September 5, 2006
- Published electronically: August 6, 2007
- Communicated by: Andreas Seeger
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3641-3648
- MSC (2000): Primary 33B15, 39B62; Secondary 26D15
- DOI: https://doi.org/10.1090/S0002-9939-07-09057-0
- MathSciNet review: 2336580