Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A half-discrete Hilbert-type inequality with a homogeneous kernel and an extension | Yang and Chen Journal of Inequalities and Applications 2011 2011 124 http content 2011 1 124 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access A half-discrete Hilbert-type inequality with a homogeneous kernel and an extension Bicheng Yang1 and Qiang Chen2 Correspondence bcyang@gdei. department of Mathematics Guangdong University of Education Guangzhou Guangdong 510303 P. R. China Full list of author information is available at the end of the article Abstract Using the way of weight functions and the technique of real analysis a half-discrete Hilbert-type inequality with a general homogeneous kernel is obtained and a best extension with two interval variables is given. The equivalent forms the operator expressions the reverses and some particular cases are considered. 2000 Mathematics Subject Classification 26D15 47A07. Keywords Hilbert-type inequality homogeneous kernel weight function equivalent form reverse 1 Introduction Assuming that p r . i f p jT Y x dxp 1 p q 1 f 0 e Ư R g 0 e Lq R 0 11 g q 0 we have the following Hardy-Hilbert s integral inequality 1 0 0 n X X f y dxdy x y sin n p f p 1 n where the constant factor sin n p is the best possible. If am bn 0 Ỉ b bn Xi lq b bn Xi lq X zl p 0 b L 0 then we still n n i n n i a p am 0 q have the following discrete Hardy-Hilbert s inequality with the same best constant fac- n tor sin n p XX ambn m n m i n i llallpllbllq. sin n p 2 Springer For p q 2 the above two inequalities reduce to the famous Hilbert s inequalities. Inequalities 1 and 2 are important in analysis and its applications 2-4 . In 1998 by introducing an independent parameter l e 0 1 Yang 5 gave an extension of 1 for p q 2. Refinement and generalizing the results from 5 Yang 6 gave some best extensions of 1 and 2 as follows If l1 l2 e R l1 l2 l ki x 2011 Yang and Chen licensee Springer. This is an Open Access article distributed under the terms of the Creative .
