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 Differences of Weighted Mixed Symmetric Means and Related Results | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 289730 16 pages doi 2010 289730 Research Article Differences of Weighted Mixed Symmetric Means and Related Results Khuram Ali Khan 1 J. PeCariC 1 2 and I. PeriC3 1 Abdus Salam School of Mathematical Sciences GC University 68-B New Muslim Town Lahore 54600 Pakistan 2 Faculty of Textile Technology University of Zagreb Pierotti-jeva 6 10000 Zagreb Croatia 3 Faculty of Food Technology and Biotechnology University of Zagreb 10002 Zagreb Croatia Correspondence should be addressed to Khuram Ali Khan khuramsms@ Received 22 June 2010 Accepted 13 October 2010 Academic Editor Marta Garcia-Huidobro Copyright 2010 Khuram Ali Khan 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. Some improvements of classical Jensen s inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined and their monotonicity is established as an application. 1. Introduction and Preliminary Results For n e N let x x1z. xn and p p1 . pn be positive n-tuples such that 1. Pi 1. We define power means of order r e R as follows Mr x p Mr x1 . Xn P1 . . . Pn r 0 r 0. We introduce the mixed symmetric means with positive weights as follows 2 Journal of Inequalities and Applications M1 t x p k k zx lMis xil j 1 1 s . . xik pi1 . . .pik yk Avck 1 H1 i1 ik n Mt Xi1 . Xik Pi1 . Pik 1 p s 0 s 0. We obtain the monotonicity of these means as a consequence of the following improvement of Jensen s inequality 1 . Theorem . Let I c R x x1 . xn e In p p1 . pn be a positive n-tuple such that 1 pi 1. Also let f I R be a convex function and 1k fk n x p C-1 s Ẹp f Ck-1 1 i1 - ik n j 1 j 1 .
