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Research Article Some New Hilbert’s Type Inequalities | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 851360 10 pages doi 10.1155 2009 851360 Research Article Some New Hilbert s Type Inequalities Chang-Jian Zhao1 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 Chang-Jian Zhao chjzhao@163.com Received 25 December 2008 Accepted 24 April 2009 Recommended by Peter Pang Some new inequalities similar to Hilbert s type inequality involving series of nonnegative terms are established. Copyright 2009 C.-J. Zhao and W.-S. 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 In recent years several authors 1-10 have given considerable attention to Hilbert s type inequalities and their various generalizations. In particular in 1 Pachpatte proved some new inequalities similar to Hilbert s inequality 11 page 226 involving series of nonnegative terms. The main purpose of this paper is to establish their general forms. 2. Main Results In 1 Pachpatte established the following inequality involving series of nonnegative terms. Theorem A. Let p 1 q 1 and let am and bn be two nonnegative sequences of real numbers defined for m 1 . k and n 1 . r where k r are natural numbers. Let Am m 1 as and Bn in 1 bt. Then k r Ap Bq k ix2 1 2 EEm n-C p V k AE k-m 1KamApm m 1 n 1 m 1 r a 2 r - n 1 bnBn-1 . n 1 X 2.1 2 Journal of Inequalities and Applications where 1 Cfy q k r 2 pq kr 1 2. 2.2 We first establish the following general form of inequality 2.1 . Theorem 2.1. Let p 1 q 1 t 0 and 1 a 1 f 1 a 1. Let am1 . mn and bn1 .n be positive sequences of real numbers defined for mi 1 2 . ki and n-i 1 2 . n where ki ri i 1 .