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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 Hilbert’s Type Linear Operator and Some Extensions of Hilbert’s Inequality | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 82138 10 pages doi 10.1155 2007 82138 Research Article Hilbert s Type Linear Operator and Some Extensions of Hilbert s Inequality Yongjin Li Zhiping Wang and Bing He Received 17 April 2007 Accepted 3 October 2007 Recommended by Ram N. Mohapatra The norm of a Hilbert s type linear operator T L2 0 to - L2 0 to is given. As applications a new generalizations of Hilbert integral inequality and the result of series analogues are given correspondingly. Copyright 2007 Yongjin Li et al. 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 At the close of the 19th century a theorem of great elegance and simplicity was discovered by D. Hilbert. Theorem 1.1 Hilbert s double series theorem . The series to to ỵỵaamanL 1.1 m n m 1 n 1 is convergent whenever TO 1an is convergent. The Hilbert s inequalities were studied extensively refinements generalizations and numerous variants appeared in the literature see 1 2 . Firstly we will recall some Hilbert s inequalities. If f x g x 0 0 J0to f 2 x dx to and 0 vg2 x dx to then 1 2 z f f x g y dxdy n f f2 x dxl i f g2 x dx 0 0 x y 0 0 1.2 where the constant factor n is the best possible. Inequality 1.2 is named of Hardy-Hilbert s integral inequality see 3 . Under the same condition of 1.2 we have the 2 Journal of Inequalities and Applications Hardy-Hilbert s type inequality see 3 Theorem 319 Theorem 341 similar to 1.2 00 00 0 0 xgy dxdy 4Ỉ f f2 x dxi f f g2 x dx max x y 0 0 1.3 where the constant factor 4 is also the best possible. The corresponding inequalities for series are 00 ambn m n n 1 m 1 7 o 0 2 o 1 2 a2n Ỉ bỤ 00 n 1 m 1 ambn max m n o X 1 2 o X 1 2 a2J d 1.4 where the constant factors n and 4 are both the best possible. Let H be a real separable Hilbert space and
