tailieunhanh - Báo cáo hóa học: " Research Article A Double Inequality for Gamma Function"

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 A Double Inequality for Gamma Function | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 503782 7 pages doi 2009 503782 Research Article A Double Inequality for Gamma Function Xiaoming Zhang1 and Yuming Chu2 1 Haining Radio and TV University Haining 314400 Zhejiang China 2 Department of Mathematics Huzhou Teachers College Huzhou 313000 Zhejiang China Correspondence should be addressed to Yuming Chu chuyuming2005@ Received 12 June 2009 Revised 21 August 2009 Accepted 30 August 2009 Recommended by Ramm Mohapatra Using the Alzer integral inequality and the elementary properties of the gamma function a double inequality for gamma function is established which is an improvement of Merkle s inequality. Copyright 2009 X. Zhang and Y. Chu. 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 psi function are defined by r x f tx 1e -dt 0 . - r x fw - r X respectively. For extensions of these functions to complex variables and for basic properties see 1 . Recently the gamma function has been the subject of intensive research many remarkable inequalities for r can be found in literature 2-21 . In particular the ratio r s r r s r 0 have attracted the attention of many mathematicians and physicists. Gautschi 22 first proved that 1-S r n t S exp 1 - s y n 1 for 0 s 1 and n 1 2 3. 2 Journal of Inequalities and Applications A strengthened upper bound was given by Erber 23 r n 1 4 n s n 1 1 s r n s 4n s 1 2 1-3 In 24 Keckic and Vasic established the following double inequality for b a 0 b ea-b r b bb- 1 2 b. aa-1 r a aa-l1 2 1-4 In 25 Kershaw obtained exp 1 - s p x s1 2 x x X S exp J1 - s q x 1 s 1Ộ x 1 2 r x s s - 4 1 21 1-s 1-5 x 1 2 for x 0 and 0 s 1. The generalized .