### KARAMATA INEQUALITY PDF

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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Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex. Belgrade in French1: By clicking “Post Inequaloty Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies.

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

Dividing by n gives Jensen’s inequaliity. Views Read Edit View history. Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

## Inequalities of Jensen and Karamata

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Convexity defined by Karamata inequality Ask Question. The finite form of Jensen’s inequality is a special case of this result. It generalizes the discrete form of Jensen’s inequality. Email Required, but never shown.

Retrieved from ” https: From Wikipedia, the free encyclopedia. By Karamata’s inequality 1 for the convex function f. 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.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold.

## Karamata’s inequality

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Sign up or log in Sign up using Google. Post as a guest Name. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?

Then x 1. Here majorization means that x 1. This is false by some stupid reasons. I am looking for a proof of weighted version of the inequality as following.

### IMOmath: Inequalities of Jensen and Karamata

Email Required, but karajata shown. Sign up using Facebook. Juanito 1, 4 Weighted version of the Karamata Inequality Ask Question. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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An explanation of Karamata’s inequality and majorization theory can be found here. Consider the real numbers x 1. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold. CS1 French-language sources fr Articles containing proofs.