Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 48727, 8 pages
doi:10.1155/JIA/2006/48727

Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions

A. Laforgia and P. Natalini

 Department of Mathematics, Roma Tre University, Largo San Leonardo Murialdo 1, Rome 00146, Italy
Received 29 June 2005; Accepted 3 July 2005

Abstract

We denote by Γ(a) and Γ(a;z) the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to x>0, a lower bound established by Elbert and Laforgia (2000) for the function ∫0xe−tpdt=[Γ(1/p)−Γ(1/p;xp)]/p, with p>1, only for 0<x<(9(3p+1)/4(2p+1))1/p.