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][2] a Romanian mathematician.

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

For a more general discussion, see Bookmaker and Odds.

Formulation

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,

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

If a function f is continuous, then it is convex if and only if the above inequality holds for all x, y, z from ${\displaystyle I}$. When f is strictly convex, the inequality is strict except for x = y = z.[3]

Generalizations

It can be generalized 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:[4]

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,

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

Popoviciu's inequality can also be generalized to a weighted inequality.[5][6][7]

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. 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
3. 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
4. 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
5. P. M. Vasić; Lj. R. Stanković (1976), "Some inequalities for convex functions", Math. Balkanica (6 (1976)), pp. 281–288
6. Grinberg, Darij (2008). "Generalizations of Popoviciu's inequality". arXiv:0803.2958v1 [math.FA].
7. M.Mihai; F.-C. Mitroi-Symeonidis (2016), "New extensions of Popoviciu's inequality", Mediterr. J. Math., Volume 13, 13 (5), pp. 3121–3133, arXiv:1507.05304, doi:10.1007/s00009-015-0675-3, ISSN 1660-5446