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Báo cáo hóa học: " Research Article On a Converse of Jensen’s Discrete Inequality"

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 Converse of Jensen’s Discrete Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 153080 6 pages doi 10.1155 2009 153080 Research Article On a Converse of Jensen s Discrete Inequality Slavko Simic Mathematical Institute SANU Kneza Mihaila 36 11000 Belgrade Serbia Correspondence should be addressed to Slavko Simic ssimic@turing.mi.sanu.ac.rs Received 10 July 2009 Revised 30 November 2009 Accepted 6 December 2009 Recommended by Martin Bohner We give the best possible global bounds for a form of discrete Jensen s inequality. By some examples the fruitfulness of this result is shown. Copyright 2009 Slavko Simic. 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 Throughout this paper x xi represents a finite sequence of real numbers belonging to a fixed closed interval I a b a b and p Pi y Pi 1 is a positive weight sequence associated with x. If f is a convex function on I then the well-known Jensen s inequality 1 2 asserts that 0 y Pif Xi - f PiXiỴ M There are many important inequalities which are particular cases of Jensen s inequality among which are the weighted A - G - H inequality Cauchy s inequality the Ky Fan and Holder s inequalities. One can see that the lower bound zero is of global nature since it does not depend on p x but only on f and the interval I whereupon f is convex. We give in 1 an upper global bound i.e. depending on f and I only which happens to be better than already existing ones. Namely we prove that 0 ỵ Pif Xi - f ỵ PiX Tf a V 1.2 2 Journal of Inequalities and Applications with Tf a b mpax pf a 1 - p f b - ffpa 1 - . 1.3 Note that for a strictly positive convex function f Jensen s inequality can also be stated in the form 1 s Pif xi f X PiXi 1.4 It is not difficult to prove that 1 is the best possible global lower bound for Jensen s inequality written in .