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 On a Multiple Hilbert-Type Integral Inequality with the Symmetric Kernel | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 27962 17 pages doi 2007 27962 Research Article On a Multiple Hilbert-Type Integral Inequality with the Symmetric Kernel Wuyi Zhong and Bicheng Yang Received 26 April 2007 Accepted 29 August 2007 Recommended by Sever S. Dragomir We build a multiple Hilbert-type integral inequality with the symmetric kernel K x y and involving an integral operator T. For this objective we introduce a norm II x n x e R two pairs of conjugate exponents p q and r s and two parameters. As applications the equivalent form the reverse forms and some particular inequalities are given. We also prove that the constant factors in the new inequalities are all the best possible. Copyright 2007 W. Zhong and B. Yang. 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 notations and lemmas If p 1 1 p 1 q 1 f x g x 0 f e Lp 0 ro g e Lq 0 ro 0 Jq00 fp x dx 1 p 00 and 0 J0TO gq y dy 1 p ro then f x g y x y dxdy ro 1 p ro 1 q fp x dx gq y dy sin n p 0 0 where the constant factor n sin n p is the best possible. Equation is the famous Hardy-Hilbert s inequality proved by Hardy-Riesz 1 in 1925. By introducing the norms II f bp llgllq and an integral operator T Lp 0 ro Lp 0 ro Yang 2 rewrite as n Tf g sm IIf II p 11g q L2 2 Journal of Inequalities and Applications where Tf g is the formal inner product of Tf and g. For f e Lp 0 to or g e Lq 0 to the integral operator T is defined by Tf y Ị0 f x x y dx or Tg x íẵ g y x y dy and II f p Jq00 I f x Ipdx 1 p llgllq J0TO g y Iq dy 1 q then to to f x to f x g y T f g dx g y dy dxdy. 0 0 x y 0 x y Inequality posts the relationship of Hilbert inequality and the integral operator T. Recently inequality has been extended by 3-6 by using the way of weight function and .
