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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 A Reverse Hardy-Hilbert-Type Inequality Gaowen Xi | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 79758 7 pages doi 10.1155 2007 79758 Research Article A Reverse Hardy-Hilbert-Type Inequality Gaowen Xi Received 12 December 2006 Accepted 12 March 2007 Recommended by Lars-Erik Persson By estimating the weight coefficient a reverse Hardy-Hilbert-type inequality is proved. As applications some equivalent forms and a number of particular cases are obtained. Copyright 2007 Gaowen Xi. 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 p 1 1 p 1 q 1 an 0 bn 0 and 0 Zn 0 an o 0 Zn 0 bỉ o. Then the Hardy-Hilbert inequality is as follows o o n 0 m 0 ambn m n 1 co 1 p co 1 q r Ĩ aP ỵ bl sin n m n 0 n 0 1.1 where the constant factor n sin np is the best possible 1 . For 1.1 Yang et al. 2-6 gave some strengthened versions and extensions as follows o o n 0 m 0 ambn m n 1 o z n 0 7 5 7 3 r 7 7 n L 5 pn 3 1.2 o o n 0 m 0 ambn m n 1 o s n 0 n sin n p ln2 - C 2n 1 1 1 p apt 1 p o ẳ n 0 n ln2 - C sin n p 2n 1 1 1 q 1 q 1.3 X n 2 Journal of Inequalities and Applications where ln2 - C 0.1159315 C is the Euler constant 00 00 n 0 m 0 ambn m n 1 A b p A 2 q A 2 p q 0 z n 0 1 1 Ả p yp n 2 an X fz n 2 Vb p Ln 0 2 J 1.4 where the constant B p A - 2 p q A - 2 q is the best possible 2 - min p q A 2 00 n 0 m 0 ambn 7---. B m n 1 A n 0 í I 1 q-1-A i 1 q 2 J 1 p 1.5 where the constant B A p A q is the best possible 0 A min p q 0 0 ambn B r - 2 t A s - 2 t Au 0 i 1 p 1-t 2t-A r -1ap t. n m n 1 A r s 2 n n 0 m 0 n 0 r z . 1 . q 1-t 2t-A s -1 i1 q 2 1.6 where the constant B r - 2 t A r s - 2 t A s is the best possible r 1 1 r 1 s 1 t G 0 1 2 - min r s t A 2 - min r s t min r s . For the reverse Hardy-Hilbert inequality recently Yang 7 gave a reverse form of inequalities 1.4 1.5 and 1.6 for A 2. The main objective of this paper
