Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 728612, 13 pages
doi:10.1155/2009/728612
Research Article

Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions

Tie-Hong Zhao,1 Yu-Ming Chu,2 and Yue-Ping Jiang3

1Institut de Mathématiques, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France
2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, Zhejiang, China
3College of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, China

Received 14 October 2008; Accepted 27 February 2009

Academic Editor: Sever Dragomir

Copyright © 2009 Tie-Hong Zhao et al. 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

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N.Guo and F.Qi. Several inequalities for the geometric means of natural numbers are established.