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 New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 256796 8 pages doi 2010 256796 Research Article On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane Zheng Zeng1 and Zitian Xie2 1 Department of Mathematics Shaoguan University Shaoguan Guangdong 512005 China 2 Department of Mathematics Zhaoqing University Zhaoqing Guangdong 526061 China Correspondence should be addressed to Zitian Xie gdzqxzt@ Received 5 May 2010 Accepted 14 July 2010 Academic Editor Andrea Laforgia Copyright 2010 Z. Zeng and Z. Xie. 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 introducing some parameters and estimating the weight functions we build a new Hilbert s inequality with the homogeneous kernel of 0 order and the integral in whole plane. The equivalent inequality and the reverse forms are considered. The best constant factor is calculated using Complex Analysis. 1. Introduction If f x g x 0 and satisfy that 0 0 f2 x dx TO and 0 0 g2 x dx TO then we have 1 Xậ ldxẠ nf r f2 x dx r g2 x dx- r 2 0 x y 0 0 J where the constant factor n is the best possible. Inequality is well known as Hilbert s integral inequality which has been extended by Hardy-Riesz as 2 . If p 1 1 p 1 q 1 f x g x 0 such that 0 0 fp x dx TO and 0 J0 gq x dx TO then we have the following Hardy-Hilbert s integral inequality n dxdy sin n p 0 gq x dx 0 where the constant factor n sin n p also is the best possible. 2 Journal of Inequalities and Applications Both of them are important in Mathematical Analysis and its applications 3 . It attracts some attention in recent years. Actually inequalities and have many generalizations and variations. Equation has been strengthened by Yang and others including double series inequalities 4-21 . In 2008 Xie and Zeng .
