tailieunhanh - Báo cáo hóa học: " Research Article On Inverse Hilbert-Type Inequalities Zhao Changjian1 and Wing-Sum Cheung2"

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 On Inverse Hilbert-Type Inequalities Zhao Changjian1 and Wing-Sum Cheung2 | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 693248 6 pages doi 2008 693248 Research Article On Inverse Hilbert-Type Inequalities Zhao Changjian1 and Wing-Sum Cheung2 1 Department of Information and Mathematics Sciences College of Science China Jiliang University Hangzhou 310018 China 2 Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong Correspondence should be addressed to Zhao Changjian chjzhao@ Received 14 November 2007 Revised 1 December 2007 Accepted 4 December 2007 Recommended by Martin J. Bohner This paper deals with new inverse-type Hilbert inequalities. Our results in special cases yield some of the recent results and provide some new estimates on such types of inequalities. Copyright 2008 Z. Changjian and . Cheung. 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 Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities and their various generalizations by several authors including Handley et al. 1 Minzhe and Bicheng 2 Minzhe 3 Hu 4 Jichang 5 Bicheng 6 and Zhao 7 8 . In 1998 Pachpatte 9 gave some new integral inequalities similar to Hilbert inequality see 10 page 226 . In 2000 Zhao and Debnath 11 established some inverse-type inequalities of the above integral inequalities. This paper deals with some new inverse-type Hilbert inequalities which provide some new estimates on such types of inequalities. 2. Main results Theorem . Let 0 pi 1 i 1 . n and r 0. Let ai mi be n positive sequences of real numbers defined for mi 1 2 . ki where ki i 1 . n are natural numbers define Am yffik. aiSi and define Ai0 0. Then for p-1 q-1 1 p 0 or 0 p 1 one has Si 1 t n n 1APmi m 1 mA 1 n sn 1mr n pr n Hpiki p i 1 v z . 1 q y ki - mi AimApmi q mi 1 2 Journal of .