tailieunhanh - Báo cáo hóa học: " Research Article On Refinements of Aczel, Popoviciu, Bellman’s ´ Inequalities and Related Results"

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 Refinements of Aczel, Popoviciu, Bellman’s ´ Inequalities and Related Results | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 579567 17 pages doi 2010 579567 Research Article On Refinements of Aczel Popoviciu Bellman s Inequalities and Related Results G. Farid 1 J. PeCariC 1 2 and Atiq Ur Rehman1 1 Abdus Salam School of Mathematical Sciences GC University 68-B New Muslim Town Lahore 54600 Pakistan 2 Faculty of Textile Technology University of Zagreb Pierottijeva 6 10000 Zagreb Croatia Correspondence should be addressed to G. Farid faridphdsms@ Received 11 July 2010 Revised 18 September 2010 Accepted 27 November 2010 Academic Editor Q. Lan Copyright 2010 G. Farid 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. We give some refinements of the inequalities of Aczel Popoviciu and Bellman. Also we give some results related to power sums. 1. Introduction The well-known Aczel s inequality 1 see also 2 page 117 is given in the following result. Theorem . Let n be a fixed positive integer and let A B ak bk k 1 . n be real numbers such that n A - Yfik 0 k 1 n B2 - b2k 0 k 1 then n V2 n A1 2 A2 - a2k B2 - b2k AB - akbk k 1 k 1 k 1 with equality if and only if the sequences A a1 . .an and B b1 . bn are proportional. A related result due to Bjelica 3 is stated in the following theorem. 2 Journal of Inequalities and Applications Theorem . Let n be a fixed positive integer and let p A B ak bk k 1 . n be nonnegative real numbers such that n n Ap - y apk 0 Bp - y bk 0 k 1 k 1 then for 0 p 2 one has 1 p n 1 p n ap - apk Bp - bk AB - akbk. k 1 k 1 k 1 Note that quotation of the above result in 4 page 58 is mistakenly stated for all p 1. In 1990 Bjelica 3 proved that the above result is true for 0 p 2. Mascioni 5 in 2002 gave the proof for 1 p 2 and gave the counter example to show that the above result is .

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