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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 New Strengthened Carleman’s Inequality and Hardy’s Inequality | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 84104 7 pages doi 10.1155 2007 84104 Research Article New Strengthened Carleman s Inequality and Hardy s Inequality Haiping Liu and Ling Zhu Received 26 July 2007 Accepted 9 November 2007 Recommended by Ram N. Mohapatra In this note new upper bounds for Carleman s inequality and Hardy s inequality are established. Copyright 2007 H. Liu and L. Zhu. 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 following Carleman s inequality and Hardy s inequality are well known. Theorem 1.1 see 1 Theorem 334 . Let an 0 n e N and 0 y TO i.an TO then TO 00 X a a2 an e an. 1.1 Theorem 1.2 see 1 Theorem 349 . Let 0 An 1 Ản An y m Am an 0 n e N and 0 y TO Anan TO then TO TO An 1 aí1 aỳ an 1 An eẢnan. n 1 n 1 1.2 In 2-16 some refined work on Carleman s inequality and Hardy s inequality had been gained. It is observing that in 3 the authors obtained the following inequalities 1 Ị 71 1 f e 1 Ị 71 -M1 2 n n 1 5 n n 1 6 1.3 2 Journal of Inequalities and Applications From the inequality above 3 4 extended Theorems A and B to the following new results. Theorem 1.3 see 3 Theorem 1 . Let an 0 n G N and 0 n 1an y then Z h i2 an 11n e 1 11 0 an- 1.4 n 1 n L Theorem 1.4 see 4 Theorem . Let 0 An 1 An An y m 1An an 0 n G N and 0 Zn 1hnan o then y y 1 -1 2 An 1 aỉ1 a an 1 An aJ 1 an- 1.5 n 1 n 1 An An 1 5 In this note Carleman s inequality and Hardy s inequality are strengthened as follows. Theorem 1.5. Let an 0 n G N 0 n 1an O and c V6 4- Then a1a2 . T Ỉ 1 - 2cn 4A 3 1 Ja- L6 n 1 n 1 Theorem 1.6. Let c ỵ 6 4 0 An 1 An An y m 1Am an 0 n G N and 0 s y 1Anan y. Then 00 y c y Id2 aAn- 2cAn 4e 3 1 2 Aj A a- 17 In order to prove two theorems mentioned above we need introduce several lemmas first. 2. Lemmas Lemma 2.1. Let x 0 and c 6
