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Research Article A Hilbert’s Inequality with a Best Constant Factor | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 820176 8 pages doi 10.1155 2009 820176 Research Article A Hilbert s Inequality with a Best Constant Factor Zheng Zeng1 and Zi-tian Xie2 1 Department of Mathematics Shaoguan University Shaoguan Guangdong 512005 China 2 Department of Mathematics Zhaoqing University Zhaoqing Guangdong 526061 China Correspondence should be addressed to Zi-tian Xie gdzqxzt@163.com Received 6 February 2009 Revised 3 May 2009 Accepted 23 July 2009 Recommended by Yong Zhou We give a new Hilbert s inequality with a best constant factor and some parameters. Copyright 2009 Z. Zeng and Z.-t. Xie. 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 If p 1 1 p 1 q 1 an bn 0 such that GO 0 1. an 0 and GO Xn 1 bn 0 then the well-known Hardy-Hilbert s inequality and its equivalent form are given by 00 00 n 1m 1 ambn m n r O ì 1 p f O Ì 1 q n ap bq sin n p tnt1 nỉ Un 1.1 O O p Vl am I i m n n 1 m 1 p sin x p 0 fan n 1 1.2 n where the constant factors are all the best possible 1 . It attracted some attention in the recent years. Actually inequalities 1.1 and 1.2 have many generalizations and variants. Equation 1.1 has been strengthened by Yang and others including integral inequalities 2-11 . 2 Journal of Inequalities and Applications In 2006 Yang gave an extension of 2 as follows. If p 1 1 p 1 q 1 r 1 1 r 1 s 1 t e 0 1 2 - min r s t min r s A 2 - min r s t such that GO 0 1. np 1-t 2t-A r -1an 0 GO Ou nq 1-t -2t-A s -1hn 0 then y amhn zim i m n A B r 2 t A s 2 t A V Vnp 1-t 2t-A r -1api pf Vnq 1-t 2t-X s -1hq q an n . n 1 n 1 1.3 B u v is the Beta function. In 2007 Xie gave a new Hilbert-type Inequality 3 as follows. If p 1 1 p 1 q 1 a h c 0 2 3 p 0 and the right of the following inequalities converges to some positive numbers then 00
