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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 New Hilbert-Type Linear Operator with a Composite Kernel and Its Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 393025 18 pages doi 10.1155 2010 393025 Research Article A New Hilbert-Type Linear Operator with a Composite Kernel and Its Applications Wuyi Zhong Department of Mathematics Guangdong Institute of Education Guangzhou Guangdong 510303 China Correspondence should be addressed to Wuyi Zhong zwy@gdei.edu.cn Received 20 April 2010 Accepted 31 October 2010 Academic Editor Ondrej Dosly Copyright 2010 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. A new Hilbert-type linear operator with a composite kernel function is built. As the applications two new more accurate operator inequalities and their equivalent forms are deduced. The constant factors in these inequalities are proved to be the best possible. 1. Introduction In 1908 Weyl 1 published the well-known Hilbert s inequality as follows if an bn 0 are real sequences 0 TOU áLn TO and 0 TOU bn TO then TO TO n 1 m 1 am bn m n TO -TO 1 2 2 a2n 2 bA 1.1 where the constant factor n is the best possible. Under the same conditions there are the classical inequalities 2 TO TO n 0 m 0 ambn m n 1 TO TO 1 2 n anỈ4 bnj ln m n ambn m - n n 1 m 1 TO TO 1 2 A taỸÁn n 1 n 1 1.2 1.3 2 Journal of Inequalities and Applications where the constant factors n and n2 are the best possible also. Expression 1.2 is called a more accurate form of 1.1 . Some more accurate inequalities were considered by 3-5 . In 2009 Zhong 5 gave a more accurate form of 1.3 . Set p q s r as two pairs of conjugate exponents and p 1 s 1 a 1 2 and an 1 1 IC rtl Af 0 w ll 4- vtp 1 x rt 1 np 3 fl n 1 w í ịy _ _ n. q 1 x s 1ljq PĨ1 if bn 0 suc L at 0 y-in 0 n a an w and 0 Zjn 0 n T a bn w l en iL has yi ln m 0 n af ambn n o m G m a x - n a x r w ì 1 p r w i 1 q n a p 1-x r -1 apn n a q 1-x s -1 bqn n 0 n 0
