Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f : ℝ^{d} \to ℝ$ that for all $x, y \in ℝ^{d}$ 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}) (\frac{\partial f}{\partial x_{i}} - \frac{\partial f}{\partial x_{j}}) \geq 0

for all

x \in 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 $x ≻ y$ 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.
A probability example: If $X_{1}, \dots, X_{n}$ are exchangeable random variables, then the function $E \prod_{j = 1}^{n} X_{j}^{a_{j}}$ 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.

References

↑ Roberts, A. Wayne; Varberg, Dale E. (1973). Convex functions. New York: Academic Press. p. 258. ISBN 9780080873725.
↑ E. Peajcariaac, Josip; L. Tong, Y. (3 June 1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226.

Schur-convex function

Properties

Schur–Ostrowski criterion

Examples

References

See also