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 to. If f is convex, then for any three points x, y, z in I,
If a function f is continuous, then it is convex if and only if the above inequality holds for all x, y, z from. 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 to. Then f is convex if and only if, for any integers n and k where n ≥ 3 and, and any n points from I,
Popoviciu's inequality can also be generalized to a weighted inequality.