Popoviciu's inequality

In convex analysis, Popoviciu's inequality is an inequality about convex functions. It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu, a Romanian mathematician.

Formulation

Let f be a function from an interval $I\subseteq \mathbb{ℝ}$ to $\mathbb{ℝ}$. If f is convex, then for any three points x, y, z in I,

$\frac{f(x)+f(y)+f(z)}{3}+f\left(\frac{x+y+z}{3}\right)\ge \frac{2}{3}[f\left(\frac{x+y}{2}\right)+f\left(\frac{y+z}{2}\right)+f\left(\frac{z+x}{2}\right)].$

If a function f is continuous, then it is convex if and only if the above inequality holds for all x, y, z from $I$. When f is strictly convex, the inequality is strict except for x = y = z.

Generalizations

It can be generalized to any finite number n of points instead of 3, taken on the right-hand side k at a time instead of 2 at a time:

Let f be a continuous function from an interval

$I\subseteq \mathbb{ℝ}$

to

$\mathbb{ℝ}$

. Then f is convex if and only if, for any integers n and k where n ≥ 3 and

$2\le k\le n-1$

, and any n points

$x_1,\dots ,x_n$

from I,

$\frac{1}{k}(\genfrac{}{}{0ex}{}{n-2}{k-2})(\frac{n-k}{k-1}\sum _{i=1}^{n}f(x_i)+nf\left(\frac{1}{n}\sum _{i=1}^n{x}_i\right))\ge \sum _{1\le i_1<\dots <i_k\le n}f\left(\frac{1}{k}\sum _{j=1}^k{x}_{i_j}\right)$

Popoviciu's inequality can also be generalized to a weighted inequality.