Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: SUPPLEMENTS TO KNOWN MONOTONICITY RESULTS AND INEQUALITIES FOR THE GAMMA AND INCOMPLETE GAMMA FUNCTIONS | SUPPLEMENTS TO KNOWN MONOTONICITY RESULTS AND INEQUALITIES FOR THE GAMMA AND INCOMPLETE GAMMA FUNCTIONS A. LAFORGIA AND P NATALINI Received 29 June 2005 Accepted 3 July 2005 We denote by r a and r 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 J0xe-tPdt r 1 p - r 1 p xp p with p 1 only for 0 x 9 3p 1 4 2p 1 1 p. 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. 1. Introduction and background In a paper of 1984 Kershaw and Laforgia 4 investigated for real a and positive x some monotonicity properties of the function xa r 1 1 x x where as usual r denotes the gamma function defined by r a e-tta-1dt a 0. 0 In particular they proved that for x 0 and a 0 the function r 1 1 x x decreases with x while when a 1 the function x r 1 1 x x increases. Moreover they also showed that the values a 0 and a 1 in the properties mentioned above cannot be improved if x E 0 to . In this paper we continue the investigation on the monotonicity properties for the gamma function proving in Section 2 the following theorem. Theorem . The functions f x r x 1 x g x r x 1 x x and h x r x 1 x decrease for 0 x 1 while increase for x 1. In Section 3 we extend a result previously established by Elbert and Laforgia 2 related to a lower bound for the integral function J0 e-tP dt with p 1. This function can be expressed by the gamma function and incomplete gamma function defined by r a z f e-tta-1dt a 0 z 0. z Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 48727 Pages 1-8 DOI JIA 2006 48727 2 Supplements to the gamma and incomplete gamma functions In fact we .
