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Báo cáo hóa học: " Research Article On a New Hilbert-Hardy-Type Integral Operator and Applications"

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 Hilbert-Hardy-Type Integral Operator and Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 812636 10 pages doi 10.1155 2010 812636 Research Article On a New Hilbert-Hardy-Type Integral Operator and Applications Xingdong Liu1 and Bicheng Yang2 1 Department of Mathematics Zhaoqing University Guangdong Zhaoqing 526061 China 2 Department of Mathematics Guangdong Institute of Education Guangdong Guangzhou 510303 China Correspondence should be addressed to Bicheng Yang bcyang@pub.guangzhou.gd.cn Received 7 September 2010 Accepted 26 October 2010 Academic Editor Sin E. Takahasi Copyright 2010 X. Liu and B. Yang. 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. By applying the way of weight functions and a Hardy s integral inequality a Hilbert-Hardy-type integral operator is defined and the norm of operator is obtained. As applications a new Hilbert-Hardy-type inequality similar to Hilbert-type integral inequality is given and two equivalent inequalities with the best constant factors as well as some particular examples are considered. 1. Introduction In 1934 Hardy published the following theorem cf. 1 Theorem 319 . Theorem A. If k x y 0 is a homogeneous function of degree -1 in 0 to X 0 to p 1 1 p 1 q 1 and kp to kfu 1 u - pdu e 0 to then for f x g y 0 0 Ilf bp ơỗ fp x dx i p TO and 0 llgllq TO one has TO 0 k x y f xig Adxdv kp f bp g llq 1.1 where the constant factor kp is the best possible. Hardy 2 also published the following Hardy s integral inequality 2 Journal of Inequalities and Applications Theorem B. If p 1 p Ỷ1 f x 0 and F x X f t dt p 1 F x x f f dt p 1 0 jX xp-p fp x dx TO then one has X pFp x dx - f xp pfp x dx 0 p- 1l 0 1.2 where the constant factor p p - 1 p is the best possible cf. 1 Theorem 330 . In 2009 Yang 3 published the following theorem. Theorem C. If p 1 1 p 1 q 1 X 0 kỵ x y 0 is a .