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 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,

${\frac {f(x)+f(y)+f(z)}{3}}+f\left({\frac {x+y+z}{3}}\right)\geq {\frac {2}{3}}\left[f\left({\frac {x+y}{2}}\right)+f\left({\frac {y+z}{2}}\right)+f\left({\frac {z+x}{2}}\right)\right].$

If a function f is continuous, then it is convex if and only if the above inequality holds for all x, y, z 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,

${\frac {1}{k}}{\binom {n-2}{k-2}}\left({\frac {n-k}{k-1}}\sum _{{i=1}}^{{n}}f(x_{i})+nf\left({\frac 1n}\sum _{{i=1}}^{{n}}x_{i}\right)\right)\geq \sum _{{1\leq i_{1}<\dots <i_{k}\leq n}}f\left({\frac 1k}\sum _{{j=1}}^{{k}}x_{{i_{j}}}\right)$

Popoviciu's inequality can also be generalised to a weighted inequality.^[4][5] Popoviciu's paper has been published in Romanian language, but the interested reader can find his results in the review Zbl 0166.06303. Page 1 Page 2

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)), pp. 281–288
5. Grinberg, Darij (2008). "Generalizations of Popoviciu's inequality". arXiv:0803.2958v1 [math.FA].