KARAMATA INEQUALITY PDF


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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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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. An explanation of Karamata’s inequality and majorization theory can be found here.

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

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Karamata’s inequality

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Belgrade in French1: Sign up or log in Sign up using Google. This is false by some stupid reasons. Sign up using Email and Password. Views Read Edit View history. 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.

Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. 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. Retrieved from ” https: Consider the real numbers x 1.

By using this site, you agree to the Terms of Use and Privacy Policy. The finite form of Jensen’s inequality is a special case of this result. Convexity defined by Karamata inequality Ask Question.

Sign up using Email and Password. Here majorization means that x 1.

IMOmath: Inequalities of Jensen and Karamata

From Wikipedia, the free encyclopedia. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms kxramata Service. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I.

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Then x 1. This page was last edited on 6 Octoberat Email Required, but never shown.

However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. Juanito 1, 4 Post inequalitg a guest Name.

Sign up using Facebook. Dividing by n gives Jensen’s inequaoity. 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.

Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Weighted version of the Karamata Inequality Ask Question. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?

From both sides of the inequality, but not from your conditions! It generalizes the discrete form of Jensen’s inequality. Email Required, but never shown. By Karamata’s inequality 1 for the convex function f .