Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality . Hence, if a function satisfies Karamata’s inequality, it satisfies. PDF | Three classical general inequalitiesâ€”those of Karamata, Schur and Muirheadâ€”are proved in this article. They can be used in proving other inequali- ties. jorization inequality where the majorization condition is replaced by a more The inequality presented in the article is a consequence of Karamata’s majoriza-.

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In mathematicsKaramata’s inequality[1] named after Jovan Karamata[2] also known as the majorization inequalityis 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. 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.

### convex analysis – Convexity defined by Karamata inequality – Mathematics Stack Exchange

Consider the real numbers x 1. Then x 1. By Karamata’s inequality 1 for the convex function f.

Dividing by n gives Jensen’s inequality. We may assume that the numbers are in decreasing order as specified in 2. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. An explanation of Karamata’s inequality and majorization theory can be found here.

From Wikipedia, the free karamat. Belgrade in French1: Retrieved from ” https: CS1 French-language sources fr Articles containing proofs. Views Read Edit View history. This page was last edited on 6 Octoberat By using this site, you agree to the Terms of Use and Privacy Policy.