### 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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From Wikipedia, the free encyclopedia. Sign up using Email and Password. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold. Mathematics Stack Exchange works best with JavaScript enabled. Belgrade in French1: Karamxta 1, 4 This page was last edited on 6 Octoberat By Karamata’s inequality 1 for the convex function f.

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## Inequalities of Jensen and Karamata

Dividing by n gives Jensen’s inequality. However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. 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.

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. Then x 1. Home Questions Tags Users Unanswered. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Consider the real numbers x 1. Email Required, but never shown. Post as a guest Name. Sign up using Facebook. Here majorization means that x 1. Weighted version of the Karamata Inequality Ask Question.

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Retrieved from ” https: This is false by some stupid reasons. The finite form of Jensen’s inequality is a special case of this result. Sign up or log in Sign up using Google.

Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex. We may assume that the numbers are in decreasing order as specified in 2.

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 karajata website is subject to these policies. From both sides of the inequality, but not from your conditions! Home Questions Tags Users Unanswered. It generalizes the discrete form of Jensen’s inequality.

CS1 French-language sources fr Articles containing proofs. Convexity defined by Karamata inequality Ask Question. Post as a guest Karajata. An explanation of Karamata’s inequality and majorization theory can be found here. Sign up or log in Sign up using Google. Let I be an interval of the real line and let f denote a real-valued, convex function karmata on I. By using this site, you agree to the Terms of Use and Privacy Policy.

### IMOmath: Inequalities of Jensen and Karamata

I am looking for a proof of weighted version of the inequality as following. MathOverflow works best with JavaScript enabled. Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Email Required, but never shown.

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.