Popoviciu's inequality

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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] a Romanian mathematician. It states:

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 xyz from . When f is strictly convex, the inequality is strict except for x = y = z.[2]

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:[3]

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 generalised to a weighted inequality.[4][5]


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