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, 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 x, y, z from . When f is strictly convex, the inequality is strict except for x = y = z.
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:
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. 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
- 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
- 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
- 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
- P. M. Vasić; Lj. R. Stanković (1976), "Some inequalities for convex functions", Math. Balkanica (6 (1976)), pp. 281–288
- Grinberg, Darij (2008). "Generalizations of Popoviciu's inequality". arXiv: [math.FA].