tailieunhanh - Báo cáo hóa học: " Research Article On a New Weighted Hilbert Inequality"

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 Weighted Hilbert Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 637397 10 pages doi 2008 637397 Research Article On a New Weighted Hilbert Inequality He Leping 1 Gao Xuemei 2 3 and Gao Mingzhe2 1 Department of Mathematics and Applied Mathematics College of Mathematics and Computer Science Jishou University Hunan Jishou 416000 China 2 Department of Mathematics and Computer Science Normal College ofJishou University Hunan Jishou 416000 China 3 Department of Mathematics College of Mathematics and Computer Science Hunan Normal University Hunan Changsha 410081 China Correspondence should be addressed to Gao Mingzhe mingzhegao@ Received 13 January 2008 Accepted 18 May 2008 Recommended by Laszló Losonczi It is shown that a weighted Hilbert inequality for double series can be established by introducing a proper weight function. Thus a quite sharp result of the classical Hilbert inequality for double series is obtained. And a similar result for the Hilbert integral inequality is also proved. Some applications are considered. Copyright 2008 He Leping 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 Let an and bn be two sequences of real numbers. It is all known that the inequality ambn . 2 2 2 fcffmfni nZ nffn M n 1 m 1 m n n 1 n 1 is called Hilbert theorem for double series 1 where 2 1 an 2X1 bn and the constant factor n2 in is the best possible value. And the equality in holds if and only if an or bn is a zero-sequence. The corresponding integral form of is that W f xigu 2 f . -dxdy n2 f2 x dx g2 x dx 0 x y-----------------------------------------------0 0 ỗ where J0 f2 x dx J0 g2 x dx and the constant factor n2 in is the best possible value. Recently various improvements and extensions of and appear in 2 Journal of .