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: INVERSES OF NEW HILBERT-PACHPATTE-TYPE INEQUALITIES | INVERSES OF NEW HILBERT-PACHPATTE-TYPE INEQUALITIES CHANG-JIAN ZHAO AND WING-SUM CHEUNG Received 7 February 2006 Revised 3 June 2006 Accepted 5 June 2006 We establish a new inequality of Hilbert type for a finite double number of nonnegative sequences of real numbers and some interrelated results which are inverse and general forms of Pachpatte s and Handley s results. An integral version and some interrelated results are also obtained. These results provide some new estimates on such types of inequalities. Copyright 2006 . Zhao and . Cheung. 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 The various generalizations and sharpenings of Hilbert s double series inequality and its integral version are obtained by Pachpatte Handley et al. Gao et al. see 1 3 5 7 10 12 16-18 . An elegant survey on this kind of inequalities was provided by Yang and Rassias see 13 . Moreover Pachpatte 9 established a new Hilbert-type inequality and its integral version as follows. Theorem . Let am bn be two nonnegative sequences ofreal numbers defined for m 1 2 . k and n 1 2 . r with a0 b0 0 and let pm qn be two positive sequences of real numbers defined for m 1 2 . k n 1 2 . r where k r are natural numbers. Define Pm m 1 ps and Qn n 1 qt. Let Ộ and y be two real-valued nonnegative convex and submultiplicative functions defined on R 0 to . Then fi am y bn m n k r m 1 n 1 k w 2 1 2 M k r k - m 1 Pm T J m 1 Pm r 2 1 2 Vbn X 2 r - n 1 qnV q Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 97860 Pages 1-11 DOI JIA 2006 97860 2 Inverses of new Hilbert-Pachpatte-type inequalities where k 2 1 2 r . 2 1 2 V 1 t fi Pm fi Qn P Q 1 Pm n 1 Qn and V am am am-1 Vbn bn bn-1. Theorem . Let f e c1 0 x R g e c1 0 y R with f 0 g 0 0 and let p ơ q r be two .
