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. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions? An explanation of Karamata’s inequality and majorization theory can be found here.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. Sign up using Facebook. Home Questions Tags Users Unanswered. We may assume that the numbers are in decreasing order as specified in 2. This iaramata false by some stupid reasons. From Wikipedia, the free encyclopedia.

## Karamata’s inequality

karsmata Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex. Here majorization means that x 1. Post as a guest Name.

Retrieved from ” https: Views Read Edit View history. 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 jnequality interval of the real line.

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This page was last edited on 6 Octoberat Sign up or log in Sign up using Google. Email Required, but never shown. I am looking for a proof of weighted version of the inequality as following. Weighted version of the Karamata Inequality Ask Question.

By Karamata’s inequality 1 for the convex function f. The finite form of Jensen’s inequality is a special case of this result. Mathematics Stack Exchange works best with JavaScript enabled.

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### inequalities – Weighted version of the Karamata Inequality – MathOverflow

Dividing by n gives Jensen’s inequality. From both sides of the inequality, but not from your conditions! 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.

By using this site, you agree to the Terms of Use and Privacy Policy. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Sign up using Email and Password. 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. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. It generalizes the discrete form of Jensen’s inequality.

Juanito 1, 4 Post as a guest Name. Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy ineqkality and cookie policyand that your continued use of the website is subject to these policies. Convexity defined by Karamata inequality Ask Question.