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 Hilbert-Type Integral Inequality in the Whole Plane with the Homogeneous Kernel of Degree −2 Dongmei Xin and Bicheng Yang | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 401428 11 pages doi 2011 401428 Research Article A Hilbert-Type Integral Inequality in the Whole Plane with the Homogeneous Kernel of Degree -2 Dongmei Xin and Bicheng Yang Department of Mathematics Guangdong Education Institute Guangzhou Guangdong 510303 China Correspondence should be addressed to Dongmei Xin xdm77108@ Received 20 December 2010 Accepted 29 January 2011 Academic Editor S. Al-Homidan Copyright 2011 D. Xin 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. By applying the way of real and complex analysis and estimating the weight functions we build a new Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree -2 involving some parameters and the best constant factor. We also consider its reverse. The equivalent forms and some particular cases are obtained. 1. Introduction If f x g x 0 satisfying 0 0 f 2 x dx TO and 0 0 g2 x dx TO then we have see 1 0 f x g y x y dxdy n f 2 x dx 0 1 2 g 2 x dx where the constant factor n is the best possible. Inequality is well known as Hilbert s integral inequality which is important in analysis and in its applications 1 2 . In recent years by using the way of weight functions a number of extensions of were given by Yang 3 . Noticing that inequality is a Homogenous kernel of degree -1 in 2009 a survey of the study of Hilbert-type inequalities with the homogeneous kernels of degree negative numbers and some parameters is given by 4 . Recently some inequalities with the homogenous kernels of degree 0 and nonhomogenous kernels have been studied see 5-9 . 2 Journal of Inequalities and Applications All of the above inequalities are built in the quarter plane. Yang 10 built a new Hilbert-type
