Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 717614, 14 pages
Research Article

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

Ye Xia

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 f(x) is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I, and f(x) is strictly convex on I. Suppose that xk[a,b]I, where 0<a<b, and pk0 for k=1,,n, and suppose that k=1npk=1. Let x̄=k=1npkxk, and σ2=k=1npk(xkx̄)2. We show k=1npkf(xk)f(x̄θ1σ2), k=1npkf(xk)f(x̄θ2σ2), for suitably chosen θ1 and θ2. 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 xk's into consideration, through σ2.