# KARAMATA INEQUALITY PDF

Life[ edit ] Jovan Karamata was born in Zagreb on February 1, into a family descended from merchants based in the city of Zemun , which was then in Austria-Hungary , and now in Serbia. In Lausanne , , he finished primary school oriented towards mathematics and sciences. In the same year he enrolled at the Engineering faculty of Belgrade University and, after several years moved to the Philosophy and Mathematicians sector, where he graduated in He spent the years — in Paris , as a fellow of the Rockefeller Foundation , and in he became Assistant for Mathematics at the Faculty of Philosophy of Belgrade University. In he was elected Full Professor at the University of Geneva.

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Shazshura inequalities — Weighted version of the Karamata Inequality — MathOverflow I am looking for a proof of weighted version of the inequality as following.

Home Questions Tags Users Unanswered. MathOverflow works best with JavaScript enabled. Belgrade in French1: We may assume that the numbers are in decreasing order as specified in 2. Sign up using Email and Password. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. Here majorization means that x 1. Then x 1. Views Read Edit View history. Sign up using Facebook. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold.

Mathematics Stack Exchange works best with JavaScript enabled. Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions? By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Consider the real numbers x 1. However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. Related Posts

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## Karamata's inequality

Shazshura inequalities — Weighted version of the Karamata Inequality — MathOverflow I am looking for a proof of weighted version of the inequality as following. Home Questions Tags Users Unanswered. MathOverflow works best with JavaScript enabled. Belgrade in French1: We may assume that the numbers are in decreasing order as specified in 2. Sign up using Email and Password. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. Here majorization means that x 1.

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## Jovan Karamata

Statement of the inequality Let be an interval of the real line and let denote a real-valued, convex function defined on. Then majorizes the -tuple, since the arithmetic mean of the largest numbers of is at least as large as the arithmetic mean of all the numbers, for every. Proof of the inequality We may assume that the numbers are in decreasing order as specified in. If for all, then the inequality holds with equality, hence we may assume in the following that for at least one. If for an, then the inequality and the majorization properties and are not affected if we remove and.

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## KARAMATA INEQUALITY PDF

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