Generalizations of Popoviciu's inequality
Abstract
We establish a general criterion for the validity of inequalities of the following form: A certain convex combination of the values of a convex function at n points and of its value at a weighted mean of these n points is always greater or equal to a convex combination of the values of the function at some other weighted means of these points. Here, the left hand side contains only one weighted mean, while the right hand side may contain as many as possible, as long as there are finitely many. The weighted mean on the left hand side must have positive weights, while those on the right hand side must have nonnegative weights. The most prominent example of such kind of inequalities, Popoviciu's inequality in its most general form, follows from the general criterion. As another application, a result by Vasile Cirtoaje is sharpened.
Cite
@article{arxiv.0803.2958,
title = {Generalizations of Popoviciu's inequality},
author = {Darij Grinberg},
journal= {arXiv preprint arXiv:0803.2958},
year = {2008}
}
Comments
The subject class "functional analysis" is an approximation; I would describe the field as "elementary inequalities for convex functions". The text should be understandable to undergraduates (like I am); everything that is used and not widely known - like a version of the Karamata inequality - is explicitely proven