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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Post Your Answer Discard By clicking “Post Your 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. Belgrade in French1: Weighted version of the Karamata Inequality Ask Question.

## Inequalities of Jensen and Karamata

Convexity defined by Karamata inequality Ask Question. Post as a guest Name. Then x 1. MathOverflow works best with JavaScript enabled.

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

## Karamata’s inequality

Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?

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However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. Here majorization means that x 1. Post Your Answer Discard By clicking “Post Your 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. Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality.

### Karamata’s inequality – Wikipedia

By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy policy and inequlaity policyand that your continued use of the website is subject to these policies. It generalizes the discrete form of Jensen’s inequality. Email Required, but never shown.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The finite form of Jensen’s inequality is a special case of this result. CS1 French-language sources fr Articles containing proofs. By Karamata’s inequality 1 for the convex function f. I am looking for a proof of weighted version of the inequality as following.

Views Read Edit View history. From both sides of the inequality, but not from your conditions! By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange works best with JavaScript enabled. By using this site, you agree to the Terms of Use and Privacy Policy. Home Questions Tags Users Unanswered. Email Required, but never shown.

### IMOmath: Inequalities of Jensen and Karamata

Consider the real numbers x 1. Juanito 1, 4 Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold. We may assume that the numbers are in decreasing order as specified in 2. Sign up or log in Sign up using Google.

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This page was last edited on 6 Octoberat From Wikipedia, the free encyclopedia. Sign up using Facebook. Retrieved from ” https: Sign up using Email and Password. In mathematicsKaramata’s inequality[1] ineqiality 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.

An explanation of Karamata’s inequality and majorization theory can be found here. Sign up using Facebook. This is false by some stupid reasons. Dividing by n gives Jensen’s inequality. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I.