Popoviciu's inequality - Alchetron, the free social 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, a Romanian mathematician. It states:

Let f be a function from an interval I ⊆ R to R . If f is convex, then for any three points x, y, z in I,

f ( x ) + f ( y ) + f ( z ) 3 + f ( x + y + z 3 ) ≥ 2 3 [ f ( x + y 2 ) + f ( y + z 2 ) + f ( z + x 2 ) ] . fxyz I fxyz

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 I ⊆ R to R . Then f is convex if and only if, for any integers n and k where n ≥ 3 and 2 ≤ k ≤ n − 1 , and any n points x 1 , … , x n from I,

1 k ( n − 2 k − 2 ) ( n − k k − 1 ∑ i = 1 n f ( x i ) + n f ( 1 n ∑ i = 1 n x i ) ) ≥ ∑ 1 ≤ i 1 < ⋯ < i k ≤ n f ( 1 k ∑ j = 1 k x i j )

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

References

Popoviciu's inequality Wikipedia

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