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 $f(x)\leq f(y)$. 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

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

holds for all $1\leq i,j\leq d$.

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.
• $x\mapsto \sum _{i=1}^{d}{x_{i}^{k}},k\geq 1$ is Schur-convex if $k\geq 1$, and Schur-concave if $k\in (0,1)$.
• 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.

See also

• Quasiconvex function