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 The Hilbert-Type Integral Inequalities with a Homogeneous Kernel of −λ-Degree | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 917392 12 pages doi 2008 917392 Research Article The Hilbert-Type Integral Inequalities with a Homogeneous Kernel of -A-Degree Wuyi Zhong Department of Mathematics Guangdong Institute of Education Guangzhou Guangdong 510303 China Correspondence should be addressed to Wuyi Zhong wp@ Received 24 March 2008 Accepted 19 May 2008 Recommended by Ondrej Dosly By introducing an integral operator a norm with a weight function and two pairs of conjugate exponents we find the conditions for the Hilbert-type integral inequalities with a homogeneous kernel of -l-degree. We also prove that the constant factors in the inequalities are all the best possible. As the particular situations some new inequalities with a homogeneous kernel and their other two forms are given. We extend some previous results. Copyright 2008 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 and preliminaries If p 1 1 p 1 q 1 f x g x 0 f G L 0 to and g G Lq 0 to such that 0 iTOfp x dx i to and 0 JoTOgq y dy 1 p to then we have fdxdy n X 7 fp x dx p 7 gq y dy p x y sin n p 0 0 where the constant factor n sin n p is the best possible. Equation is the famous Hardy-Hilbert inequality proved by Hardy et al. 1 . Let K 1 x y Tf y f0 K x y f x dx and f bp Jo f x pdx 1 p or Tg x Ị0K x yìg yidy and IlgHq -0TO g y qdy 1 q . Yang 2 rewrote as . . n . . . f si - fl Ipllg lq- where T Lr 0 to Lr 0 to r p q is an integral operator Tf g J0TO J0TOK x yf x dx gfy dy ỊỊ0TO K x y f x g y dxdy Tg f is the formal inner product of Tf and g f lip or llgllq is the norm of function f in Lp 0 to or function g in Lq 0 to . 2 Journal of Inequalities and Applications If K x y is a real measurable function and satisfies K
