In mathematics , Karamata's inequality ,  named after Jovan Karamata ,  also known as the majorization inequality , is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality , and generalizes in turn to the concept of Schur-convex functions. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. Here majorization means that x 1 ,. The finite form of Jensen's inequality is a special case of this result.
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Karamata's Inequality states that if majorizes and is a convex function , then. We will first use an important fact:. This is proven by taking casework on. If , then. A similar argument shows for other values of. Now, define a sequence such that:. Define the sequences such that and similarly. Then, assuming and similarily with the 's, we get that. Now, we know:.
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If , then A similar argument shows for other values of. Now, define a sequence such that: Define the sequences such that and similarly. See also. Categories : Stubs Theorems Inequality.
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Inequalities of Jensen and Karamata