Schur-convex function

Definition

Let $n \ge 1$ . For $x \in \R^n$ , let $x_{[1]} \ge x_{[2]} \ge \dots \ge x_{[n]}$ be the entries of $x$ in decreasing order. A vector $y \in \R^n$ majorizes $x \in \R^n$ , written $x \prec y$ , if $\sum_{i=1}^{k} x_{[i]} \le \sum_{i=1}^{k} y_{[i]} \quad \text{for } k = 1, \dots, n - 1, \qquad \sum_{i=1}^{n} x_{[i]} = \sum_{i=1}^{n} y_{[i]};$ see majorization. A function $f \colon \R^n \to \R$ is Schur-convex if $f(x) \le f(y)$ for all $x, y \in \R^n$ with $x \prec y$ . It is Schur-concave if $-f$ is Schur-convex.

Remarks

A Schur-convex function is symmetric: if $y$ is obtained from $x$ by a permutation of the entries, then $x \prec y$ and $y \prec x$ , so $f(x) = f(y)$ . Every symmetric convex function is Schur-convex. For a symmetric $f$ with continuous partial derivatives on $\R^n$ , the Schur–Ostrowski criterion says that $f$ is Schur-convex if and only if $(x_i - x_j)\big(\partial_i f(x) - \partial_j f(x)\big) \ge 0$ for all $x$ and all $i, j$ .

Example: $f(x) = \max_i x_i = x_{[1]}$ . If $x \prec y$ , the case $k = 1$ of the definition gives $x_{[1]} \le y_{[1]}$ , so $f(x) \le f(y)$ .

Non-example: $f(x_1, x_2) = x_1$ on $\R^2$ . The vectors $(1, 0)$ and $(0, 1)$ majorize each other, but $f(1, 0) = 1 > 0 = f(0, 1)$ .

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Majorization concept
Permutation definition
Schur-convex function definition

Tag	`24CEMRAGJ8`
URI	`m2://24CEMRAGJ8`
Kind	definition
Writing	W0 stub
MSC	26B25, 26D15
Defines	Schur-convex function
Uses	Majorization, Permutation
Changed	2 November 2026

Schur-convex function

Definition

Remarks

Reading path

Further reading

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