Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function that for all such that is majorized by, one has that. Named after Issai Schur, Schur-convex functions are used in the study of majorization.
A function f is 'Schur-concave' if its negative, −f, is Schur-convex.

Properties

Every function that is convex and symmetric is also Schur-convex.
Every Schur-convex function is symmetric, but not necessarily convex.
If is Schur-convex and is monotonically increasing, then is Schur-convex.
If is a convex function defined on a real interval, then is Schur-convex.

Schur–Ostrowski criterion

If f is symmetric and all first partial derivatives exist, then
f is Schur-convex if and only if
holds for all.

Examples

is Schur-concave while is Schur-convex. This can be seen directly from the definition.
The Shannon entropy function is Schur-concave.
The Rényi entropy function is also Schur-concave.
is Schur-convex if, and Schur-concave if.
The function is Schur-concave, when we assume all. In the same way, all the elementary symmetric functions are Schur-concave, when.
A natural interpretation of majorization is that if then is less spread out than. So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the median absolute deviation is not.
A probability example: If are exchangeable [random variables], then the function is Schur-convex as a function of, assuming that the expectations exist.
The Gini coefficient is strictly Schur convex.