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Báo cáo hóa học: " Research Article The Best Lower Bound Depended on Two Fixed Variables for Jensen’s Inequality with Ordered Variables"

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 The Best Lower Bound Depended on Two Fixed Variables for Jensen’s Inequality with Ordered Variables | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 128258 12 pages doi 10.1155 2010 128258 Research Article The Best Lower Bound Depended on Two Fixed Variables for Jensen s Inequality with Ordered Variables Vasile Cirtoaje Department of Automatic Control and Computers University ofPloiesti 100680 Ploiesti Romania Correspondence should be addressed to Vasile Cirtoaje vcirtoaje@upg-ploiesti.ro Received 16 June 2010 Accepted 4 November 2010 Academic Editor R. N. Mohapatra Copyright 2010 Vasile Cirtoaje. 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 the best lower bound for the weighted Jensen s discrete inequality with ordered variables applied to a convex function f in the case when the lower bound depends on f weights and two given variables. Furthermore under the same conditions we give some sharp lower bounds for the weighted AM-GM inequality and AM-HM inequality. 1. Introduction Let x x1 x2 . . xn be a sequence of real numbers belonging to an interval I and let p p1 p2 . pn be a sequence of given positive weights associated to x and satisfying pi p2 pn 1. If f is a convex function on I then the well-known discrete Jensen s inequality 1 states that à f p x 0 1.1 where à f p x p1f x1 p2f x pnf xn - f px p2x2 p n 1.2 is the so-called Jensen s difference. The next refinement of Jensen s inequality was proven in 2 as a consequence of its Theorem 2.1 part ii . f p x max 1 i k n pif Xi pkf x - pi pkf pi T pk 0. 1.3 2 Journal of Inequalities and Applications By 1.3 for fixed xi and xk we get fx Pif xi Pkf xk - pi Pky PiX PkXk X Pi -r Pk SịỊ f Xi Xk . 1.4 In this paper we will establish that the best lower bound Lprf Xi xk of Jensen s difference A. f P x for x1 xi xk xn 1.5 has the expression Lp f xxk Qif xi Rkf xk - Qi Rk f Qi t RkXk 1.6 Qi Rk where Qi P1 .