Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f:{\mathbb{ℝ}}^d\to \mathbb{ℝ}$ 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

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

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

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.
• $x\mapsto \sum _{i=1}^d{x}_i^k,k\ge 1$ is Schur-convex if $k\ge 1$, 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