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Báo cáo hóa học: " Research Article Hermite-Hadamard Inequality on Time Scales"

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 Hermite-Hadamard Inequality on Time Scales | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 287947 24 pages doi 10.1155 2008 287947 Research Article Hermite-Hadamard Inequality on Time Scales Cristian Dinu Department of Mathematics University of Craiova 200585 Craiova Romania Correspondence should be addressed to Cristian Dinu c.dinu@yahoo.com Received 21 April 2008 Revised 30 June 2008 Accepted 15 August 2008 Recommended by Patricia J. Y. Wong We discuss some variants of the Hermite-Hadamard inequality for convex functions on time scales. Some improvements and applications are also included. Copyright 2008 Cristian Dinu. 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 Recently new developments of the theory and applications of dynamic derivatives on time scales were made. The study provides an unification and an extension of traditional differential and difference equations and in the same time it is a unification of the discrete theory with the continuous theory from the scientific point of view. Moreover it is a crucial tool in many computational and numerical applications. Based on the well-known A delta and V nabla dynamic derivatives a combined dynamic derivative so-called oa diamond-a dynamic derivative was introduced as a linear combination of A and V dynamic derivatives on time scales. The diamond-a dynamic derivative reduces to the A derivative for a 1 and to the V derivative for a 0. On the other hand it represents a weighted dynamic derivative on any uniformly discrete time scale when a 1 2. See 1-5 for the basic rules of calculus associated with the diamond-a dynamic derivatives. The classical Hermite-Hadamard inequality gives us an estimate from below and from above of the mean value of a convex function. The aim of this paper is to establish a full analogue of this .