Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $$f: \mathbb{R}^d\rightarrow \mathbb{R}$$ that for all $$x,y\in \mathbb{R}^d $$ such that $$x$$ is majorized by $$y$$, one has that $$f(x)\le f(y)$$. 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 (under permutations of the arguments) is also Schur-convex.

Every Schur-convex function is symmetric, but not necessarily convex.

If $$f$$ is (strictly) Schur-convex and $$g$$ is (strictly) monotonically increasing, then $$g\circ f$$ is (strictly) Schur-convex.

If $$ g $$ is a convex function defined on a real interval, then $$ \sum_{i=1}^n g(x_i) $$ 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

$$(x_i - x_j)\left(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}\right) \ge 0 $$ for all $$x \in \mathbb{R}^d$$

holds for all 1 ≤ i ≠ j ≤ d.

Examples

 * $$ f(x)=\min(x) $$ is Schur-concave while $$ f(x)=\max(x) $$ is Schur-convex. This can be seen directly from the definition.
 * The Shannon entropy function $$\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}$$ is Schur-concave.
 * The Rényi entropy function is also Schur-concave.
 * $$ \sum_{i=1}^d{x_i^k},k \ge 1 $$ is Schur-convex.
 * $$ \sum_{i=1}^d{x_i^k},0 < k < 1 $$ is Schur-concave.
 * The function $$ f(x) = \prod_{i=1}^d x_i $$ is Schur-concave, when we assume all $$ x_i > 0 $$. In the same way, all the elementary symmetric functions are Schur-concave, when $$ x_i > 0 $$.
 * A natural interpretation of majorization is that if $$ x \succ y $$ then $$ x $$ is less spread out than $$ y $$. 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 $$ X_1, \dots, X_n $$ are exchangeable random variables, then the function $$  \text{E} \prod_{j=1}^n X_j^{a_j} $$ is Schur-convex as a function of $$ a=(a_1, \dots, a_n) $$, assuming that the expectations exist.
 * The Gini coefficient is strictly Schur convex.