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A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
Journal of Inequalities and Applications volume 2008, Article number: 717614 (2008)
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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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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