Popoviciu's inequality

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

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

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,

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

^{[5] [6] [7] [8]}

Weighted inequality

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

Let f be a continuous function from an interval ${\displaystyle I\subseteq \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$. Let ${\displaystyle x_{1},x_{2},x_{3}}$ be three points from ${\displaystyle I}$, and let ${\displaystyle w_{1},w_{2},w_{3}}$ be three nonnegative reals such that ${\displaystyle w_{2}+w_{3}\neq 0,w_{3}+w_{1}\neq 0}$ and ${\displaystyle w_{1}+w_{2}\neq 0}$. Then,

${\displaystyle {\begin{aligned}&w_{1}f\left(x_{1}\right)+w_{2}f\left(x_{2}\right)+w_{3}f\left(x_{3}\right)+\left(w_{1}+w_{2}+w_{3}\right)f\left({\frac {w_{1}x_{1}+w_{2}x_{2}+w_{3}x_{3}}{w_{1}+w_{2}+w_{3}}}\right)\\&\geq \left(w_{2}+w_{3}\right)f\left({\frac {w_{2}x_{2}+w_{3}x_{3}}{w_{2}+w_{3}}}\right)+\left(w_{3}+w_{1}\right)f\left({\frac {w_{3}x_{3}+w_{1}x_{1}}{w_{3}+w_{1}}}\right)+\left(w_{1}+w_{2}\right)f\left({\frac {w_{1}x_{1}+w_{2}x_{2}}{w_{1}+w_{2}}}\right)\end{aligned}}}$

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, 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)