Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 717614, 14 pages
A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611-6120, USA
Received 23 January 2008; Accepted 9 May 2008
Academic Editor: Ondrej Dosly
Copyright © 2008 Ye Xia. 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.
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 .