Schur-concave function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f: \mathbb^d\rightarrow \mathbb$ that for all $x,y\in \mathbb^d$ 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 $\sum_^n g(x_i)$ 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
: $(x_i - x_j)\left(\frac - \frac\right) \ge 0$ for all $x \in \mathbb^d$
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 $\sum_^d$ is Schur-concave.
* The Rényi entropy function is also Schur-concave.
* $x \mapsto \sum_^d,k \ge 1$ is Schur-convex if $k \geq 1$, and Schur-concave if $k \in (0, 1)$.
* The function $f(x) = \prod_^d x_i$ 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 $\text \prod_^n X_j^$ 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

See also

* Quasiconvex function

Convex analysis
Inequalities (mathematics)

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