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 Diamond-α Jensen’s Inequality on Time Scales | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 576876 13 pages doi 2008 576876 Research Article Diamond-a Jensen s Inequality on Time Scales Moulay Rchid Sidi Ammi Rui A. C. Ferreira and Delfim F. M. Torres Department of Mathematics University ofAveiro 3810-193 Aveiro Portugal Correspondence should be addressed to Delfim F. M. Torres delfim@ Received 12 December 2007 Revised 10 February 2008 Accepted 7 April 2008 Recommended by Martin Bohner The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-a derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen s inequality on time scales via the diamond-a integral and present some corollaries including Holder s and Minkowski s diamond-a integral inequalities. Copyright 2008 Moulay Rchid Sidi Ammi 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 Jensen s inequality is of great interest in the theory of differential and difference equations as well as other areas of mathematics. The original Jensen s inequality can be stated as follows. Theorem see 1 . If g G C fa b c d and f G C c d R is convex then rtng s ds tf g s ds A a . b a b a Jensen s inequality on time scales via A-integral has been recently obtained by Agarwal Bohner and Peterson. Theorem see 2 . If g G Crd a b c d and f G C c d R is convex then g s As tf g s As a o J a f b a b a Under similar hypotheses we may replace the A-integral by the V-integral and get a completely analogous result 3 . The aim of this paper is to extend Jensen s inequality to an arbitrary time scale via the diamond-a .
