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,[1][2] a Romanian mathematician.
Let f be a function from an interval
⊆
{\displaystyle I\subseteq \mathbb {R} }
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
{\displaystyle I}
When f is strictly convex, the inequality is strict except for x = y = z.
[3] 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:[4] Let f be a continuous function from an interval
Then f is convex if and only if, for any integers n and k where n ≥ 3 and
, and any n points
from I, [5][6][7][8] Popoviciu's inequality can also be generalized to a weighted inequality.
[9] Let f be a continuous function from an interval
be three points from
be three nonnegative reals such that