Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel | Wang and Yang Journal of Inequalities and Applications 2011 2011 123 http content 2011 1 123 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel Aizhen Wang and Bicheng Yang Correspondence zhenmaths@ Department of Mathematics Guangdong University of Education Guangzhou 510303 Guangdong People s Republic of China Springer Abstract By using the way of weight functions and the technique of real analysis a new Hilbert-type integral inequality with the non-homogeneous kernel in the whole plane with the best constant factor is given. As applications the equivalent inequalities with the best constant factors the reverses and some particular cases are obtained. 2000 Mathematics Subject Classification 26D15 Keywords weight function Hilbert-type integral inequality non-homogeneous kernel equivalent form 1 Introduction If f x g x 0 such that 0 f f 2 x dx X and 0 Ỉ0 g2 x áx X then we have cf. 1 1 ị Ị f xgyidxdy n ịf2 x dxịg2 x dxj 1 0 0 0 0 where the constant factor n is the best possible. Inequality 1 is well known as Hilbert s integral inequality which is important in Mathematical Analysis and its applications 2 . If p r 1 p q 1 r 7 1l 0 f x g x 0 such that 0 0 1 r fp x dx X and 0 0 xq 1 7 g x dx X then we have 3 X X ị ị min x y 2f x g y dxdy 0 0 1 1 2 X p X q Ị xp 1 r -1fp x dxl lị xq 1 7 -1gq x dxl 0 o where the constant factor y is the best possible. By using the way of weight functions we can get two Hilbert-type integral inequalities with non-homogeneous kernels similar to 1 and 2 as follows 4 5 2011 Wang and Yang licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. .
