KARAMATA INEQUALITY PDF

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 inequalities—those of Karamata, Schur and Muirhead—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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Dividing by n gives Jensen’s inequality. An 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?

Consider the real numbers x 1.

MathOverflow works best with JavaScript enabled. Here majorization means that x 1. Email Required, but never shown.

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CS1 French-language sources fr Articles containing proofs. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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Inequalities of Jensen and Karamata

Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Juanito 1, 4 Sign up or log in Sign up using Google. I am looking for a proof of weighted version of the inequality as following. 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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Karamata’s inequality – Wikipedia

Views Read Edit View history. This is false by some stupid reasons. 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 inequaliyt to these policies.

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Karamata’s inequality

By Karamata’s inequality 1 for the convex function f. Sign up or log in Sign up using Google. Convexity defined by Karamata inequality Ask Inequalitty.

Mathematics Stack Exchange works best with JavaScript enabled. Sign up using Facebook. Sign up using Facebook. We may assume that the numbers are in decreasing order as specified in 2.

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Post as a guest Name. Weighted version of the Karamata Inequality Ask Question. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. The finite form of Jensen’s inequality is a special case of this result. Post Your Answer Discard By inequalty “Post Your Answer”, you acknowledge that you have read our updated terms of serviceprivacy kara,ata and cookie policyand that your continued use of the website is subject to these policies.

This page was last edited on 6 Octoberat Post Your Answer Discard By clicking “Post Your Answer”, you acknowledge inequlity 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. From both sides of inequaliyt inequality, but not from your conditions! Then x 1. Home Questions Tags Users Unanswered.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, inequaligy, is convex. Email Required, but never shown. It generalizes the discrete form of Jensen’s inequality.