tailieunhanh - Báo cáo hóa học: " Research Article A Hilbert-Type Linear Operator with the Norm and Its Applications"

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 Hilbert-Type Linear Operator with the Norm and Its Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 494257 18 pages doi 2009 494257 Research Article A Hilbert-Type Linear Operator with the Norm and Its Applications Wuyi Zhong Department of Mathematics Guangdong Institute of Education Guangzhou Guangdong 510303 China Correspondence should be addressed to Wuyi Zhong wp@ Received 9 February 2009 Accepted 9 March 2009 Recommended by Nikolaos Papageorgiou A Hilbert-type linear operator T ểp fp is defined. As for applications a more precise operator inequality with the norm and its equivalent forms are deduced. Moreover three equivalent reverses from them are given as well. The constant factors in these inequalities are proved to be the best possible. Copyright 2009 Wuyi Zhong. 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 1925 Hardy 1 extended Hilbert inequality as follows. If p 1 1 p 1 q 1 an bn 0 0 yTO an TO and 0 y TO 1 bqn TO then n n TO TO n 1 m 1 ambn m n n sin n p TO ẳ n 1 V an TO 1 P 1 q TO ẳ s n 1 Lm 1 TP am n fy ap sin n pJ 21 n m n where p q is a pair of conjugate exponents. The constant factors n sin n p and n sin n p P are the best possible. The expression is the famous Hardy-Hilbert s inequality. 2 Journal of Inequalities and Applications Under the same conditions there are the classic inequalities 2 ln m n amb n 2 1 p 7 - ẵẫ m-7 liMn rtJ J 9 y y ln m n am p m - n n 1 m 1 J n yap sin n p J à where the constant factors n sin n p 2 and n sin n p 2p are also the best possible. The expression is well known as a Hilbert-type inequality. By setting a real space of sequences p a a an 0 ahp 1 an p 1 p and defining a linear operator T Ta n Cn y 1 ln m n am m - n n e N0 the expressions and can be rewritten as Ta b TlinanpnbHq Ta p .