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
Department of Mathematics, Roma Tre University, Largo San Leonardo Murialdo 1, Rome 00146, Italy
Received 29 June 2005; Accepted 3 July 2005
Copyright © 2006 A. Laforgia and P. Natalini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.