A Refinement of Jensen's Inequality for a Class of Increasing and Concave Functions
© Ye Xia. 2008
Received: 23 January 2008
Accepted: 9 May 2008
Published: 22 May 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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