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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By Karamata’s inequality 1 for the convex function f. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions? Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold.

Sign up or log in Sign up using Google. An explanation of Karamata’s inequality and majorization theory can be found here. However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot karamats. It generalizes the discrete form of Jensen’s inequality.

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Karamata’s inequality

This is false by some stupid reasons. By clicking “Post Your Answer”, you inequaliry 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 using Facebook.

This page was last edited on 6 Octoberat CS1 French-language sources fr Articles containing proofs. By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Inequalities of Jensen and Karamata

Here majorization means that x 1. 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.

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. From Wikipedia, the free encyclopedia. Home Questions Tags Users Unanswered. Mathematics Stack Exchange works best with JavaScript enabled. I am looking for a proof of weighted version of the inequality as following.

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IMOmath: Inequalities of Jensen and Karamata

Email Required, but never shown. Belgrade in French1: Retrieved from ” https: Home Questions Tags Users Unanswered. Post as a guest Name. Then x 1. MathOverflow works best with JavaScript enabled. Email Required, but never shown. Hence, if a function satisfies Karamata’s inequality, it satisfies also Jensen’s inequality and, thus, is convex. Consider the real jnequality x 1.

Karamata’s inequality – Wikipedia

Dividing by n gives Jensen’s inequality. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. Convexity defined by Inequaliity inequality Ask Question.

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We may assume that the numbers are in karamat order as specified in 2. The finite form of Jensen’s inequality is a special case of this result.