Popoviciu's inequality should not 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] a Romanian mathematician.
Let f be a function from an interval
I\subseteqR
R
f(x)+f(y)+f(z) | |
3 |
+f\left(
x+y+z | |
3 |
\right)\ge
2 | |
3 |
\left[f\left(
x+y | |
2 |
\right)+f\left(
y+z | |
2 |
\right)+f\left(
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
I
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:
Let f be a continuous function from an interval[2]toI\subseteqR
. Then f is convex if and only if, for any integers n and k where n ≥ 3 andR
, and any n points2\leqk\leqn-1
from I,x1,...,xn
1 k \binom{n-2}{k-2}\left(
n-k k-1
n \sum i=1 f(xi)+nf\left(
1n\sum i=1 nxi\right)\right)\ge
\sum 1\lei1<...<ik\len f\left(
1k \sum
k j=1
x ij \right)
Popoviciu's inequality can also be generalized to a weighted inequality.
Let f be a continuous function from an interval
I\subseteqR
R
x1,x2,x3
I
w1,w2,w3
w2+w3\ne0,w3+w1\ne0
w1+w2\ne0
\begin{aligned}&w1f\left(x1\right)+w2f\left(x2\right)+w3f\left(x3\right)+\left(w1+w2+w3\right)f\left(
w1x1+w2x2+w3x3 | |
w1+w2+w3 |
\right)\\&\geq\left(w2+w3\right)f\left(
w2x2+w3x3 | |
w2+w3 |
\right)+\left(w3+w1\right)f\left(
w3x3+w1x1 | |
w3+w1 |
\right)+\left(w1+w2\right)f\left(
w1x1+w2x2 | |
w1+w2 |
\right)\end{aligned}