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 Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 130958 12 pages doi 2009 130958 Research Article Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel Predrag Vukovic Faculty of Teacher Education University of Zagreb Savska cesta 77 10000 Zagreb Croatia Correspondence should be addressed to Predrag Vukovic Received 11 July 2009 Revised 10 November 2009 Accepted 18 November 2009 Recommended by Radu Precup We consider the Hilbert-type inequalities with nonconjugate parameters. The obtaining of the best possible constants in the case of nonconjugate parameters remains still open. Our generalization will include a general homogeneous kernel. Also we obtain the best possible constants in the case of conjugate parameters when the parameters satisfy appropriate conditions. We also compare our results with some known results. Copyright 2009 Predrag Vukovic. 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 Let 1 p 1 q 1 p 1 f g 0 0 fp x dx TO 0 gq x dx TO. 0 0 The well-known Hardy-Hilbert s integral inequality see 1 is given by Í ÍTO f w y dxdy X if x dx p f A x dx 0 x y sin n p 0 0 and an equivalent form is given by Í 7 TT dxỴdy . n . 1 i fp x dx J0V0 x y p sin x p .V where the constant factors n sin n p and n sin n p p are the best possible. 2 Journal of Inequalities and Applications During the previous decades the Hilbert-type inequalities were discussed by many authors who either reproved them using various techniques or applied and generalized them in many different ways. For example we refer to a paper of Yang see 2 . If n e N 1 pi 1 1X1 1 pi 1 s 0 fi 0 satisfy 0 xP s Ỵ fp x dx TO i 1 2 . n 0 then f nn f x t 1 s TO 1 Pi i 1fi Xi dx . fa 1 r s xPi-s- 1fPi x dx s L4x1 uxn x fi x x j xj T s
