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: Letter to the Editor Remarks on “On a Converse of Jensen’s Discrete Inequality” of S. Simic ´ | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 309565 4 pages doi 2011 309565 Letter to the Editor Remarks on On a Converse of Jensen s Discrete Inequality of S. Simic S. Ivelic1 and J. PeCariC2 1 Faculty of Civil Engineering and Architecture University of Split Matice Hrvatske 15 21000 Split Croatia 2 Faculty of Textile Technology University of Zagreb Prilaz Baruna Filipovica 30 10000 Zagreb Croatia Correspondence should be addressed to S. Ivelic sivelic@ Received 13 January 2011 Accepted 10 February 2011 Copyright 2011 S. Ivelic and J. Pecaric. 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 show that the main results by S. Simic are special cases of results published many years earlier by J. E. Pecaric et al. 1992 . Let I be an interval in R and ộ I R a convex function on I .If x X1 . xn is any n-tuple in In and p p1 . . pn a positive n-tuple such that n 1 pi 1 then the well known Jensen s inequality ộ y Pix y piộ xi Ơ holds see . 1 page 43 . If ộ is strictly convex then 1 is strict unless xi c for all i e j Pj 0 . The following results are given in 2 . Theorem 1. Let I a b where a b x x1z. . xn e In and p p1 . . pn Xn 1 pi 1 be a sequence of positive weights associated with x. Let ộ be a strictly positive twice continuously differentiable function on I and 0 p q 1 p q 1. One has that i if ộ is a strictly convex function on I then 1 jXfffff max ộ n 1 Pixi p pộa qộ ff 1 r Sộ M ộ pa qb 2 2 Journal of Inequalities and Applications ii if Ộ is a strictly concave function on I then 1 ft s 1 PiXi ệ j a qb Pi M i pa P a qộ b S ộ a b . 3 Both estimates are independent of p. Theorem 2. There is unique P0 e 0 1 such that S tn h - P0 a 1- P W ệ a b Ộ P0U 1 -P0 b . 4 The main aim of our paper is to show that the main results presented in 2 .
