A simple proof of Popoviciu's inequality

Abstract

T. Popoviciu [5] has proved in 1965 the following inequality relating the values of a convex function $f:I\to \mathbb{R}$ at the weighted arithmetic means of the subfamilies of a given family of points $x_1,...,x_n\in I$:$$\begin{array}{rl}& \sum _{1\le i_1<\cdots <i_p\le n}({\lambda }_{i_1}+\cdots +{\lambda }_{i_p})f\left(\frac{{\lambda }_{i_1}{x}_{i_1}+\cdots +{\lambda }_{i_p}{x}_{i_p}}{{\lambda }_{i_1}+\cdots +{\lambda }_{i_p}}\right)\\ & \le (\genfrac{}{}{0ex}{}{n-2}{p-2})[\frac{n-p}{p-1}\sum _{i=1}^n{\lambda }_{i}f(x_i)+\left(\sum _{i=1}^n{\lambda }_i\right)f\left(\frac{{\lambda }_1{x}_1+\cdots +{\lambda }_n{x}_n}{{\lambda }_1+\cdots +{\lambda }_n}\right)].\end{array}$$Here $n\ge 3,$ $p\in \{2,...,n-1\}$ and ${\lambda }_1,...,{\lambda }_n$ are positive numbers (representing weights). The aim of this paper is to give a simple argument based on mathematical induction and a majorization lemma.