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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In mathematicsKaramata’s inequality named after Jovan Karamata 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.
It generalizes the discrete form of Jensen’s inequality. 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. The finite form of Jensen’s inequality is a special case of this result.
convex analysis – Convexity defined by Karamata inequality – Mathematics Stack Exchange
Consider the real numbers x 1. Then x 1. By Karamata’s inequality 1 for the convex function f.
Dividing by n gives Jensen’s inequality. We may assume that the numbers are in decreasing order as specified in 2. Hence there is a strictly positive term in the sum on the right hand side of 7 and equality in 1 cannot hold.
However, then there is a strictly positive term on the right hand side of 7 and equality in 1 cannot hold. An explanation of Karamata’s inequality and majorization theory can be found here.