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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### IMOmath: Inequalities of Jensen and Karamata

Sign up using Email and Password. Email Required, but never shown. Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Post as a guest Name. From Wikipedia, the free encyclopedia. Belgrade in French1: Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex.

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Sign up using Facebook. By Karamata’s inequality 1 for the convex function f. MathOverflow works best with JavaScript enabled.

### Karamata’s inequality – Wikipedia

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By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The finite form of Jensen’s inequality is a special case of this result. Home Questions Tags Users Unanswered. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold.

Weighted version of the Karamata Inequality Ask Question. 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 an interval of the real line. Retrieved from ” https: Home Questions Tags Users Unanswered.

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This is false by some stupid reasons. It generalizes the discrete form of Jensen’s inequality.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold.

Convexity defined by Karamata inequality Ask Question. Sign up using Email and Password.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. CS1 French-language sources fr Articles containing proofs. Dividing by n gives Jensen’s inequality. I am looking for a proof of weighted version of the inequality karramata following.

## Inequalities of Jensen and Karamata

Here majorization means that x 1. Consider the real numbers x 1. Then x 1. Mathematics Stack Exchange works best with JavaScript enabled. Sign up using Facebook. By using this site, you agree to the Terms of Use and Privacy Policy.

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## Karamata’s inequality

Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. This page was last edited on 6 Octoberat From both sides of the inequality, but not from your conditions!

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