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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## Inequalities of Jensen and Karamata

Views Read Edit View history. By using our site, inequaliy acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

This page was last edited on 6 Octoberat 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. By Karamata’s inequality 1 for the convex function f. Hence, if a function satisfies Karamata’s inequality, it satisfies ibequality Jensen’s inequality and, thus, is convex.

Dividing by n gives Jensen’s inequality.

Post as a guest Name. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold. This is false by some stupid reasons.

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. Sign up or log in Sign up using Google. However, then there is a strictly positive term on the right hand side of 7 and equality karramata 1 cannot hold.

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

Email Required, but never shown. From Wikipedia, the free encyclopedia. Sign up using Email and Password. Retrieved from ” https: I am looking for a proof of weighted version of the inequality as following. Wikipedia says that Jensen’s inequality is a special case kaeamata Karamata’s inequality.

By clicking “Post Your Answer”, you acknowledge that you have read our updated terms inequaliyt serviceprivacy policy and cookie policyand that your continued use of the website is subject to these policies. An explanation of Karamata’s inequality and majorization theory can be found here. The finite form of Jensen’s inequality is a special case of this result.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. It generalizes the discrete form of Jensen’s inequality.

Email Required, but never shown. 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.

By using this site, you agree to the Terms of Use and Privacy Policy. Belgrade in French1: Here majorization means that x 1. MathOverflow works best with JavaScript enabled.

### Karamata’s inequality – Wikipedia

Mathematics Stack Exchange works iinequality with JavaScript enabled. Home Questions Tags Users Unanswered. Weighted version of the Karamata Inequality Ask Question. Sign up using Email and Password. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I.

Sign up using Facebook. 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.

Then x 1. Post as a guest Name. Juanito 1, 4 Just karammata the Jensen inequality is used karamsta define convex functions, can the Karamata inequality be used instead to define convex functions?

Convexity defined by Karamata inequality Ask Question.