Popoviciu's inequality

From Wikipedia the free encyclopedia

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.

Formulation

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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.[3]

Generalizations

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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 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,

[5][6][7][8]

Weighted inequality

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Popoviciu's inequality can also be generalized to a weighted inequality.[9]

Let f be a continuous function from an interval to . Let be three points from , and let be three nonnegative reals such that and . Then,

Notes

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  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, no. 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, vol. 13, no. 5, pp. 3121–3133, arXiv:1507.05304, doi:10.1007/s00009-015-0675-3, ISSN 1660-5446, S2CID 119720352
  8. ^ M.W. Alomari (2021), "Popoviciu's type inequalities for h-MN-convex functions", e-Journal of Analysis and Applied Mathematics, Volume 2021, vol. 2021, no. 1, pp. 48–89, doi:10.2478/ejaam-2021-0005
  9. ^ Darij Grinberg, Generalizations of Popoviciu’s inequality (PDF)