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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 Jensen’s Inequality for Convex-Concave Antisymmetric Functions and Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 185089 6 pages doi 10.1155 2008 185089 Research Article Jensen s Inequality for Convex-Concave Antisymmetric Functions and Applications S. Hussain 1 J. Pecaric 1 2 and I. Peric3 1 Abdus Salam School of Mathematical Sciences GC University Lahore Gulberge Lahore 54660 Pakistan 2 Faculty of Textile Technology University of Zagreb 10000 Zagreb Croatia 3 Faculty of Food Technology and Biotechnology University of Zagreb Pierottijeva 6 10000 Zagreb Croatia Correspondence should be addressed to S. Hussain sabirhus@gmail.com Received 21 February 2008 Accepted 9 September 2008 Recommended by Lars-Erik Persson The weighted Jensen inequality for convex-concave antisymmetric functions is proved and some applications are given. Copyright 2008 S. Hussain et al. 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 The famous Jensen inequality states that f p ini 1 1 n E Pif x 1.1 where f I R is a convex function I is interval in R xi e I pi 0 i 1 . n and Pn y n 1 pi. Recall that a function f I R is convex if f 1 - f x tỳ 1 - i f x if y 1.2 holds for every x y e I and every t e 0 1 see 1 Chapter 2 . The natural problem in this context is to deduce Jensen-type inequality weakening some of the above assumptions. The classical case is the case of Jensen-convex or mid-convex functions. A function f I R is Jensen-convex if f x y f x f y 1.3 2 Journal of Inequalities and Applications holds for every x y e I. It is clear that every convex function is Jensen-convex. To see that the class of convex functions is a proper subclass of Jensen-convex functions see 2 page 96 . Jensen s inequality for Jensen-convex functions states that if f I R is a Jensen-convex function then f 1 nx 1 Yf x 1.4 f n2-xi n2-iJ W 4 1 1 ni 1 where xi e
