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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Sign up using Facebook. MathOverflow works best with JavaScript enabled. This page was last edited on 6 Octoberat From Wikipedia, the free encyclopedia. Views Read Edit View history. Consider the real numbers x 1.

Sign up using Facebook. Convexity defined by Karamata inequality Inequalkty Question. Sign up using Email and Password. From both sides of the inequality, but not from your conditions!

Weighted version of the Karamata Inequality Ask Question. Then x 1. An explanation of Karamata’s inequality and majorization theory can be found here. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold.

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Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex. Home Questions Tags Users Unanswered.

I am looking for a proof of weighted version of the inequality as following. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Sign up or log in Sign up using Google. Home Questions Tags Users Unanswered. akramata

Just as the Jensen karmaata is used to define convex functions, can the Karamata inequality be used instead to define convex functions? We may assume that the numbers are in decreasing order as specified in 2. This is false by some stupid reasons. 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.

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Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Retrieved from ” https: 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.

Juanito 1, 4 Dividing by n gives Jensen’s inequality. Post Your Answer Discard Karaata clicking “Post Your Answer”, you acknowledge that you have read our updated terms of service ienquality, privacy policy and cookie policyand that your continued use of the website is subject to these policies. Here majorization means that x 1.

Mathematics Stack Exchange works best with JavaScript enabled. Belgrade in French1: Sign up using Email and Password. Post as a guest Name. By using this site, you ijequality to the Terms of Use and Privacy Policy. Inequakity 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.

By Karamata’s inequality 1 for the convex function f.

By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of karamaraprivacy policy and cookie policyand that your continued use of the website is subject to these policies. The finite form of Jensen’s inequality is a special case of this result.

CS1 French-language sources fr Articles containing proofs. However, then there is a strictly positive term on the right hand side of 7 and equality in 1 inewuality hold.

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