JIPAM

Inequalities for the Gamma Function  
 
  Authors: Xin Li, Chao-Ping Chen,  
  Keywords: Gamma function, psi function, inequality.  
  Date Received: 16/10/06  
  Date Accepted: 09/02/07  
  Subject Codes:

Primary 33B15; Secondary 26D07.

  
  Editors: Andrea Laforgia,  
 
  Abstract:

For $ x>1$, the inequalities

$\ displaystyle $\ frac{x^{x-$\ gamma}}{e^{x-1}}<$\ Gamma(x)<$\ frac{x^{x-1/2}}{e^{x-1}} $
hold, and the constants $ $\ gamma$ and $ 1/2$ are the best possible, where $ $\ gamma=0.577215$\ ldots$ is the Euler-Mascheroni constant. For $ 0<x<1$, the left-hand inequality also holds, but the right-hand inequality is reversed. This improves the result given by G. D. Anderson and S. -L. Qiu (1997). ;


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