Popoviciu's inequality

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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 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 xyz from . 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 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.[4][5] [6]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


  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].
  6. ^ M.Mihai; F.-C. Mitroi-Symeonidis (2016), "New extensions of Popoviciu's inequality", Mediterr. J. Math., Volume 13 (5), pp. 3121–3133, doi:10.1007/s00009-015-0675-3, ISSN 1660-5446