Numerical Algorithms

, Volume 49, Issue 1–4, pp 53–84 | Cite as

Gamma function inequalities

  • Horst AlzerEmail author
Original Paper


We prove various new inequalities for Euler’s gamma function. One of our theorems states that the double-inequality
$$\alpha \cdot \Bigl(\frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr) {\kern-1pt}<{\kern-1pt} \frac{1}{\Gamma\,( \sqrt{x+y-xy})}+ \frac{1}{\Gamma\,( \sqrt{xy})} {\kern-1pt} <{\kern-1pt} \beta \cdot \Bigl( \frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr)$$
is valid for all real numbers x,y ∈ (0,1) with the best possible constant factors \(\alpha=1/\sqrt{2}=0.707...\) and β = 1.


Gamma function Inequalities Monotonicity Convexity Mean values 

Mathematics Subject Classification (2000)



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Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.WaldbrölGermany

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