Open Access

A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions

Journal of Inequalities and Applications20082008:717614

DOI: 10.1155/2008/717614

Received: 23 January 2008

Accepted: 9 May 2008

Published: 22 May 2008

Abstract

Suppose that is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval , and is strictly convex on . Suppose that , where , and for , and suppose that . Let , and . We show , , for suitably chosen and . These results can be viewed as a refinement of the Jensen's inequality for the class of functions specified above. Or they can be viewed as a generalization of a refined arithmetic mean-geometric mean inequality introduced by Cartwright and Field in 1978. The strength of the above result is in bringing the variations of the 's into consideration, through .

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Authors’ Affiliations

(1)
Department of Computer and Information Science and Engineering, University of Florida

Copyright

© Ye Xia. 2008

This article is published under license to BioMed Central Ltd. 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.