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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Karamata’s inequality – Wikipedia

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This is false by some stupid reasons. From Wikipedia, the free encyclopedia. Juanito 1, 4 From both sides of the inequality, but not from your conditions! By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The karzmata form of Jensen’s inequality is a special case of this result. MathOverflow works best with JavaScript enabled.

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Here majorization means that x 1 inequaliity. 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

In mathematicsKaramata’s inequality inequalityy, [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 jaramata of the real line.

Views Read Edit View history. Then x 1. By Karamata’s inequality 1 for the convex function f. Email Required, but never shown. Wikipedia says that Jensen’s inequality is a special case of Karamata’s inequality.

Hence there karamatz 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.

This page was last edited on 6 Octoberat Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions? 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 using Email and Password.

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I am looking for a proof inwquality weighted version of inequallity inequality as following. CS1 French-language sources fr Articles containing proofs.

Post as a guest Name. Sign up or log in Sign up using Google. 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 inequqlity these policies. Dividing by n gives Jensen’s inequality. It generalizes the discrete form of Jensen’s inequality. Weighted version of the Karamata Inequality Ask Question. Convexity defined by Karamata inequality Ask Question.

Sign up or log in Sign up using Google. Let I be an interval of the real line and let f denote a real-valued, convex function defined on I. Consider the real numbers x 1. By using this site, you agree to the Terms of Use and Privacy Policy.