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 Some New Results Related to Favard’s Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 128486 14 pages doi 2009 128486 Research Article Some New Results Related to Favard s Inequality Naveed Latif 1 J. Pecaric 1 2 and I. Peric3 1 Abdus Salam School of Mathematical Sciences GC University Lahore 54000 Pakistan 2 Faculty of Textile Technology University of Zagreb 10000 Zagreb Croatia 3 Faculty of Food Technology and Biotechnology University of Zagreb 10000 Zagreb Croatia Correspondence should be addressed to Naveed Latif sincerehumtum@ Received 31 July 2008 Revised 17 January 2009 Accepted 5 February 2009 Recommended by A. Laforgia Log-convexity of Favard s difference is proved and Drescher s and Lyapunov s type inequalities for this difference are deduced. The weighted case is also considered. Related Cauchy type means are defined and some basic properties are given. Copyright 2009 Naveed Latif 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 and p be two positive measurable real valued functions defined on a b Q R with ịbap x dx 1. From theory of convex means cf. 1 2 the well-known Jensen s inequality gives that for t 0 or t 1 p x fi x dx p x f x dx a a and reverse inequality holds for 0 t 1. In 3 Simic considered the difference Ds Ds a b f p p x fs x dx - p x f x dn . aa He has given the following. 2 Journal of Inequalities and Applications Theorem . Let f and p be nonnegative and integrable functions on a b with Jbap x dx 1 then for 0 r s t r s t 1 one has Ds t-r s s -1 s Dr t-s Dt s-r r r - 1 t t - 1 Remark . For an extension of Theorem see 3 . Let us write the well-known Favard s inequality. Theorem . Let f be a concave nonnegative function on a b c R. If q 1 then b fq x dx. a If 0 q 1 the reverse .
