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 Iterative Refinements of the Hermite-Hadamard Inequality, Applications to the Standard Means | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 107950 13 pages doi 2010 107950 Research Article Iterative Refinements of the Hermite-Hadamard Inequality Applications to the Standard Means Sever S. Dragomir1 2 and Mustapha Raissouli3 1 Research Group in Mathematical Inequalities and Applications School of Engineering and Science Victoria University . Box 14428 Melbourne City MC 8001 Australia 2 School of Computational and Applied Mathematics University of the Witwatersrand Private Bag 3 Wits 2050 Johannesburg 2000 South Africa 3 Applied Functional Analysis Team AFACSI Laboratory Faculty of Science Moulay Ismaỉl University . Box 11201 Meknes Morocco Correspondence should be addressed to Mustapha Ra issouli raissoulL10@ Received 29 July 2010 Accepted 19 October 2010 Academic Editor Laszlo A. Losonczi Copyright 2010 S. S. Dragomir and M. Ra issouli. 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. Two adjacent recursive processes converging to the mean value of a real-valued convex function are given. Refinements of the Hermite-Hadamard inequality are obtained. Some applications to the special means are discussed. A brief extension for convex mappings with variables in a linear space is also provided. 1. Introduction Let C be a nonempty convex subset of R and let f C R be a convex function. For a b e C the following double inequality a b 1 b f a f b f 2 - b- aiaf x dx - 2 1 is known in the literature as the Hermite-Hadamard inequality for convex functions. Such inequality is very useful in many mathematical contexts and contributes as a tool for establishing some interesting estimations. In recent few years many authors have been interested to give some refinements and extensions of the Hermite-Hadamard inequality 1-4 . Dragomir 1 .
