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 An Extension of the Hilbert’s Integral Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 158690 12 pages doi 2009 158690 Research Article An Extension of the Hilbert s Integral Inequality Zhou Yu Gao Xuemei and Gao Mingzhe Department of Mathematics and Computer Science Normal College ofJishou University Hunan Jishou 416000 China Correspondence should be addressed to Gao Mingzhe mingzhegao@ Received 30 May 2009 Accepted 20 August 2009 Recommended by Wing-Sum Cheung It is shown that an extension of the Hilbert s integral inequality can be established by introducing two parameters m m e N and Ằ A- 0 . The constant factors expressed by the Euler number and n as well as by the Bernoulli number and n respectively are proved to be the best possible. Some important and especial results are enumerated. As applications some equivalent forms are given. Copyright 2009 Zhou Yu et al. 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 Lemmas Let f x g x e L2 0 x . Define ln x y 0 1 when x y. If k 0 1 then ln x y kf x g y 1 1 ki1 n 1 2 2 1 2 z_x k------dxdy nk 1 f2 x dx g2 x dx 0 x -1 ky 0 0 where the constant factor nk 1 is the best possible. This is the famous Hilbert s integral inequality see 1 2 . Owing to the importance of the Hilbert s inequality and the Hilbert-type inequality in analysis and applications some mathematicians have been studying them. Recently various improvements and extensions of appear in a great deal of papers see 3-11 etc. . Specially Gao and Hsu enumerated the research articles more than 40 in the paper 6 . The purpose of the present paper is to establish the Hilbert-type inequality of the form ln x yY nf x g y x _ 1 2 x 2 1 2 z_ ---- z -------dxdy cl Ũ x f2 x dx t Ũ x g2 x dxt 0 x -1 V 0 0 2 Journal of Inequalities and Applications where n is
