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 \(f:I\rightarrow\mathbb{R}\) at the weighted arithmetic means of the subfamilies of a given family of points \(x_{1},...,x_{n}\in I\):\begin{align*}& \sum\limits_{1\leq i_{1}<\cdots <i_{p}\leq n}(\lambda_{i_{1}}+\cdots +\lambda _{i_{p}})\,f\left( \tfrac{\lambda_{i_{1}}x_{i_{1}}+\cdots +\lambda_{i_{p}}x_{i_{p}}}{\lambda _{i_{1}}+\cdots +\lambda _{i_{p}}}\right) \\& \leq \tbinom{n-2}{p-2}\left[\tfrac{n-p}{p-1}\,\sum\limits_{i=1}^{n}\,\lambda_{i}\,f(x_{i})+\left( \sum\limits_{i=1}^{n}\,\lambda _{i}\right) \,f\left( \tfrac{\lambda _{1}x_{1}+\cdots +\lambda _{n}x_{n}}{\lambda _{1}+\cdots+\lambda _{n}}\right) \right] .\end{align*}Here \(n\geq 3,\) \(p\in \{2,...,n-1\}\) 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.