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. It states:

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

It can be generalised 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 ${\displaystyle I\subseteq \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$. Then f is convex if and only if, for any integers n and k where n ≥ 3 and ${\displaystyle 2\leq k\leq n-1}$, and any n points ${\displaystyle x_{1},\dots ,x_{n}}$ from I,

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