Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f:\mathbb {R} ^{d}\rightarrow \mathbb {R}$ , for which if $\forall x,y\in \mathbb {R} ^{d}$ where $x$ is majorized by $y$ , then $f(x)\leq f(y)$ . Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).

Schur-concave function

A function $f$ is 'Schur-concave' if its negative, $-f$ , 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

$(x_{i}-x_{j})({\frac {\partial f}{\partial x_{i}}}-{\frac {\partial f}{\partial x_{j}}})\geq 0$ for all $x\in \mathbb {R} ^{d}$

holds for all 1≤i≠j≤d.^[1]

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.

$\sum _{i=1}^{d}{x_{i}^{k}},k\geq 1$ is Schur-convex.

The function $f(x)=\prod _{i=1}^{n}x_{i}$ 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 more 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.

If $g$ is a convex function defined on a real interval, then $\sum _{i=1}^{n}g(x_{i})$ is Schur-convex.

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.