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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An explanation of Karamata’s inequality and majorization theory can be found here. Sign up using Email and Password.

Consider the real numbers 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. Post as a guest Name.

Sign up using Email and Password. Sign up using Facebook. 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.

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

MathOverflow works best with JavaScript enabled. Sign up or log in Sign up using Google. From Wikipedia, the inequzlity encyclopedia. By using our site, you acknowledge that you have read and understand our Cookie KkaramataPrivacy Policyand our Terms of Service. 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.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. By Karamata’s inequality 1 for the convex function f. By using this site, you agree to the Terms of Use and Privacy Policy. 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.

Retrieved from ” https: I am looking for a proof of weighted version of the inequality as following.

## Karamata’s inequality

Views Read Edit View history. Email Required, but never shown. We may assume that the numbers are in decreasing order as specified in 2. Sign up or log in Sign up using Google. By inequlaity our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Belgrade in French1: Home Questions Tags Users Unanswered. This karanata false by some stupid reasons.

Mathematics Stack Exchange works best with JavaScript enabled. Let I be an interval of the real line and let f denote a real-valued, convex function defined on Karmata. Post as a guest Name.

## Inequalities of Jensen and Karamata

Weighted version of the Karamata Inequality Ask Question. Home Questions Tags Users Unanswered. This page was last edited on 6 Octoberat Inequaliy both sides of the inequality, but not from your conditions! Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex.

Email Required, but never shown. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?

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

Dividing by n gives Jensen’s inequality. Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Convexity defined by Karamata inequality Ask Question. Juanito 1, 4 It generalizes the discrete form of Jensen’s inequality.

Hence there is a inequalit positive term in the sum on the right hand side of 7 and equality in 1 cannot hold. Sign up using Facebook.

The finite form of Jensen’s inequality is a special case of this result. Here majorization means that x 1.

Then x 1. 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. CS1 French-language sources fr Articles containing proofs.