tailieunhanh - Báo cáo hóa học: " Research Article Cauchy Means of the Popoviciu Type"

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 Cauchy Means of the Popoviciu Type | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 628051 16 pages doi 2009 628051 Research Article Cauchy Means of the Popoviciu Type Matloob Anwar 1 Naveed Latif 1 and J. PeCariC1 2 1 Abdus Salam School of Mathematical Sciences 68-B New Muslim Town GC University Lahore 54000 Pakistan 2 Faculty of Textile Technology University of Zagreb Pierottijeva 6 Zagreb 10000 Croatia Correspondence should be addressed to Naveed Latif sincerehumtum@ Received 9 October 2008 Accepted 2 February 2009 Recommended by Wing-Sum Cheung We discuss log-convexity for the differences of the Popoviciu inequalities and introduce some mean value theorems and related results. Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic. Copyright 2009 Matloob Anwar 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 and Preliminaries Let f x and p x be two positive real valued functions with Jap x dx 1 then from theory of convex means cf. 1-3 the well-known Jensen inequality gives that for t 0 or t 1 p x f x ỹdx p x f x dx a J a and vise versa for 0 t 1. In 4 Simic has considered the difference Ds Ds a b f p p x f x sdx - p x f x dx J a a The following result was given in 4 see also 5 . 2 Journal of Inequalities and Applications Theorem . Let f x p x be nonnegative and integrable functions for x e a b with Ja p x dx 1 then for 0 r s t r s t 1 one has Ds t-r s s - 1 2 Dr t-s Dt s-r r r - 1 t t - 1 Remark . For extension of Theorem see cf. 4 . Popoviciu 6-8 9 pages 214-215 has proved the following results. Theorem . Let Ộ a b R be convex and f 0 1 R be continuous increasing and convex such that a f x b for xe 0 1 . Then where .1 ộ f xỴ dx 0 b a 2 ộ a zt-a. f bộ x dx b - a r b - a M a f f x dx.