 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-. Author: Zushicage Arar Country: Guinea Language: English (Spanish) Genre: Relationship Published (Last): 14 September 2010 Pages: 317 PDF File Size: 5.77 Mb ePub File Size: 11.64 Mb ISBN: 223-6-70884-881-1 Downloads: 67120 Price: Free* [*Free Regsitration Required] Uploader: Bagul 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.

COLLIN MOSHMAN SNG STRATEGY PDF 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. 