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

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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Correspondence to Ye Xia.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Xia, Y. A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions. J Inequal Appl 2008, 717614 (2008). https://doi.org/10.1155/2008/717614

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  • Concave Function
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