tailieunhanh - Báo cáo sinh học: " Research Article Improvement and Reversion of Slater’s Inequality and Related Results"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Improvement and Reversion of Slater’s Inequality and Related Results | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 646034 14 pages doi 2010 646034 Research Article Improvement and Reversion of Slater s Inequality and Related Results M. Adil Khan1 and J. E. Pecaric1 2 1 Abdus Salam School of Mathematical Sciences GC University Lahore 54000 Pakistan 2 Faculty of Textile Technology University of Zagreb Zagreb 10002 Croatia Correspondence should be addressed to M. Adil Khan adilbandai@ Received 6 March 2010 Accepted 2 June 2010 Academic Editor Kunquan Lan Copyright 2010 M. Adil Khan and J. E. 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 use an inequality given by Matic and Pecaric 2000 and obtain improvement and reverse of Slater s and related inequalities. 1. Introduction In 1981 Slater has proved an interesting companion inequality to Jensen s inequality 1 . Theorem . Suppose that ộ I c R R is increasing convex function on interval I for x1 x2 . xn e I where I is the interior of the interval I and for p1 p2 . pn 0 with Pn En 1 pi 0 if En 1 Piộ Xi 0 then 1 ưzn 1 Piộ Xi Xi P Pộ Xi s ộ V. pộ x When ộ is strictly convex on I inequality becomes equality if and only if xi c for some c e I and for all i with Pi 0. It was noted in 2 that by using the same proof the following generalization of Slater s inequality 1981 can be given. 2 Journal of Inequalities and Applications Theorem . Suppose that ộ I c R R is convex function on interval I for x1 x2 . xn e I where I is the interior of the interval I and for p1 p2 . pn 0 with Pn n i pi 0- Let s 1 piộ .xi xi TO 2ỷiộ x f0 ff e I Ơ-2 i 1 2-ii 1 piộ xi then inequality holds- When ộ is strictly convex on I inequality becomes equality if and only if xi c for some c e IO and for all i with pi 0- Remark 1-3- For multidimensional version of .