Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function [math]\displaystyle{ f: \mathbb{R}^d\rightarrow \mathbb{R} }[/math] that for all [math]\displaystyle{ x,y\in \mathbb{R}^d }[/math] such that [math]\displaystyle{ x }[/math] is majorized by [math]\displaystyle{ y }[/math], one has that [math]\displaystyle{ f(x)\le f(y) }[/math]. 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 [math]\displaystyle{ f }[/math] is (strictly) Schur-convex and [math]\displaystyle{ g }[/math] is (strictly) monotonically increasing, then [math]\displaystyle{ g\circ f }[/math] is (strictly) Schur-convex.

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

[math]\displaystyle{ (x_i - x_j)\left(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}\right) \ge 0 }[/math] for all [math]\displaystyle{ x \in \mathbb{R}^d }[/math]

holds for all 1 ≤ i ≠ j ≤ d.

Examples

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

See also

• Quasiconvex function