Popoviciu's inequality

Not to be confused with Popoviciu's inequality on variances.

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 $I \subseteq \mathbb{R}$ to $\mathbb{R}$. If f is convex, then for any three points x, y, z in I,

$\frac{f(x)+f(y)+f(z)}{3} + f\left(\frac{x+y+z}{3}\right) \ge \frac{2}{3}\left[ f\left(\frac{x+y}{2}\right) + f\left(\frac{y+z}{2}\right) + f\left(\frac{z+x}{2}\right) \right].$
If a function f is continuous, then it is convex if and only if the above inequality holds for all xyz from $I$. 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 $I \subseteq \mathbb{R}$ to $\mathbb{R}$. Then f 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,

$\frac{1}{k} \binom{n-2}{k-2} \left( \frac{n-k}{k-1} \sum_{i=1}^{n}f(x_i) + nf\left(\frac1n\sum_{i=1}^{n}x_i\right) \right)\ge \sum_{1 \le i_1 < \dots < i_k \le n} f\left( \frac1k \sum_{j=1}^{k} x_{i_j} \right)$

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].