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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 A Refinement of Jensen’s Inequality for a Class of Increasing and Concave Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 717614 14 pages doi 10.1155 2008 717614 Research Article A Refinement of Jensen s Inequality for a Class of Increasing and Concave Functions Ye Xia Department of Computer and Information Science and Engineering University of Florida Gainesville FL 32611-6120 USA Correspondence should be addressed to Ye Xia yx1@cise.ufl.edu Received 23 January 2008 Accepted 9 May 2008 Recommended by Ondrej Dosly Suppose that f x is strictly increasing strictly concave and twice continuously differentiable on a nonempty interval I-- and f- x is strictly convex on I. Suppose that xk- a b -I where 0 a b and pk-0 for k 1 - n and suppose that -n 1pk 1. Let x _n 1pkxk and _2 -n 1pk xk_x 2. We show -n 1pkf xk _f x __1_2 -n 1pkf xk _f x __2_2 for suitably chosen _1 and _2. These results can be viewed as a refinement of the Jensen s inequality for the class of functions specified above. Or they can be viewed as a generalization of a refined arithmetic mean-geometric mean inequality introduced by Cartwright and Field in 1978. The strength of the above result is in bringing the variations of the xk s into consideration through _2. Copyright 2008 Ye Xia. 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. Main theorem The goal is to generalize the following refinement of the arithmetic mean-geometric mean inequality introduced in 1 . The result in this paper can also be viewed as a refinement of Jensen s inequality for a class of increasing and concave functions. Many other refinements can be found in 2 . Theorem 1.1 see 1 . Suppose that xk a b and pk 0for k 1 . n where a 0 and suppose that n iPk 1- Then writing x n iPkxk 1 n 9 n 1 n 2b yPk xk - x 2 x - Hxpk 2a yPkx - x 2. 1.1 For notational simplicity define Ơ2 yPk xk - x 2 2pkxk - x. 1.2 k
