# 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,[1] a Romanian mathematician. It states:

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

<\math> \begin{align} & {} \qquad \frac{f(x)+f(y)+f(z)}{3} + f\left(\frac{x+y+z}{3}\right) \\[6pt] & \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]. \end{align} [/itex]

If a function ƒ is continuous, then it is convex if and only if the above inequality holds for all xyz from $I$. When ƒ 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 ƒ be a continuous function from an interval $I \subseteq \mathbb{R}$ to $\mathbb{R}$. Then ƒ is convex if and only if, for any integers n and k where n ≥ 3 and $2 \leq k \leq n-1$, and any $n$ points $x_1, \dots, x_n$ from $I$,

[/itex] \begin{align} & {} \qquad \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)\\[6pt] & \ge \sum_{1 \le i_1 < \dots < i_k \le n} f\left( \frac1k \sum_{j=1}^{k} x_{i_j} \right) \end{align} [/itex]

Popoviciu's inequality can also be generalised to a weighted inequality.[4][5]

## Notes

1. ^ Tiberiu Popoviciu (1965), "Sur certaines inégalités qui caractérisent les fonctions convexes", Analele ştiinţifice Univ. "Al.I. Cuza" Iasi, Secţia I a Mat. 11: 155–164
2. ^ Constantin Niculescu; Lars Erik Persson (2006), Convex functions and their applications: a contemporary approach, Springer Science & Business, p. 12, ISBN 978-0-387-24300-9
3. ^ J. E. Pečarić; Frank Proschan; Yung Liang Tong (1992), Convex functions, partial orderings, and statistical applications, Academic Press, p. 171, ISBN 978-0-12-549250-8
4. ^ P. M. Vasić; Lj. R. Stanković (1976), "Some inequalities for convex functions", Math. Balkanica (6 (1976)): 281–288
5. ^ Grinberg, Darij (2008). "Generalizations of Popoviciu's inequality". arXiv:0803.2958v1 [math.FA].