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: Research Article Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 728612 13 pages doi 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 1 Institut de Mathematiques Université Pierre et Marie Curie 4 Place Jussieu 75252 Paris France 2 Department of Mathematics Huzhou Teachers College Huzhou 313000 Zhejiang China 3 College of Mathematics and Econometrics Hunan University Changsha 410082 Hunan China Correspondence should be addressed to Yu-Ming Chu chuyuming2005@ Received 14 October 2008 Accepted 27 February 2009 Recommended by Sever Dragomir 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 . Guo and F. Qi. Several inequalities for the geometric means of natural numbers are established. 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. 1. Introduction For real and positive values of x the Euler gamma function r and its logarithmic derivative y the so-called digamma function are defined as to r x tx-1e dt 0 - r x r X For extension of these functions to complex variables and for basic properties see 1 . In recent years many monotonicity results and inequalities involving the Gamma and incomplete Gamma functions have been established. This article is stimulated by an open problem posed by Guo and Qi in 2 . The extensions and generalizations of this problem can be found in 3-5 and some references therein. Using Stirling formula for all nonnegative integers k natural numbers n and m an upper bound of the quotient of two
