Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sub- and superadditive properties of Euler's gamma function


Author: Horst Alzer
Journal: Proc. Amer. Math. Soc. 135 (2007), 3641-3648
MSC (2000): Primary 33B15, 39B62; Secondary 26D15
Published electronically: August 6, 2007
MathSciNet review: 2336580
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33B15, 39B62, 26D15

Retrieve articles in all journals with MSC (2000): 33B15, 39B62, 26D15


Additional Information

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

DOI: http://dx.doi.org/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
Published electronically: August 6, 2007
Communicated by: Andreas Seeger
Article copyright: © Copyright 2007 American Mathematical Society