Generalizations of Popoviciu's inequality

TL;DR

This paper develops a broad criterion for inequalities involving convex functions and weighted means, unifying and extending classical results like Popoviciu's inequality and sharpening existing inequalities.

Contribution

It introduces a general criterion that determines when certain convex inequalities involving weighted means hold, encompassing Popoviciu's inequality as a special case.

Findings

Established a unifying criterion for convex inequalities involving weighted means

Generalized Popoviciu's inequality to a broader class of inequalities

Sharpened a known inequality by Vasile Cirtoaje

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.