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 [math]\displaystyle{ I \subseteq \mathbb{R} }[/math] to [math]\displaystyle{ \mathbb{R} }[/math]. If f is convex, then for any three points x, y, z in I,

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

If a function f is continuous, then it is convex if and only if the above inequality holds for all x, y, z from [math]\displaystyle{ I }[/math]. 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 [math]\displaystyle{ I \subseteq \mathbb{R} }[/math] to [math]\displaystyle{ \mathbb{R} }[/math]. Then f is convex if and only if, for any integers n and k where n ≥ 3 and [math]\displaystyle{ 2 \leq k \leq n-1 }[/math], and any n points [math]\displaystyle{ x_1, \dots, x_n }[/math] from I,

[math]\displaystyle{ \frac{1}{k} \binom{n-2}{k-2} \left( \frac{n-k}{k-1} \sum_{i=1}^{n}f(x_i) + nf\left(\frac1n\sum_{i=1}^{n}x_i\right) \right)\ge \sum_{1 \le i_1 \lt \dots \lt i_k \le n} f\left( \frac1k \sum_{j=1}^{k} x_{i_j} \right) }[/math]

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