In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function ${\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} }$ that for all ${\displaystyle x,y\in \mathbb {R} ^{d}}$ such that ${\displaystyle x}$ is majorized by ${\displaystyle y}$, one has that ${\displaystyle f(x)\leq f(y)}$. 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. ^[1]

If ${\displaystyle f}$ is (strictly) Schur-convex and ${\displaystyle g}$ is (strictly) monotonically increasing, then ${\displaystyle g\circ f}$ is (strictly) Schur-convex.

If ${\displaystyle g}$ is a convex function defined on a real interval, then ${\displaystyle \sum _{i=1}^{n}g(x_{i})}$ 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

${\displaystyle (x_{i}-x_{j})\left({\frac {\partial f}{\partial x_{i}}}-{\frac {\partial f}{\partial x_{j}}}\right)\geq 0}$ for all ${\displaystyle x\in \mathbb {R} ^{d}}$

holds for all 1 ≤ i ≠ j ≤ d. ^[2]

Examples

• ${\displaystyle f(x)=\min(x)}$ is Schur-concave while ${\displaystyle f(x)=\max(x)}$ is Schur-convex. This can be seen directly from the definition.
• The Shannon entropy function ${\displaystyle \sum _{i=1}^{d}{P_{i}\cdot \log _{2}{\frac {1}{P_{i}}}}}$ is Schur-concave.
• The Rényi entropy function is also Schur-concave.
• ${\displaystyle \sum _{i=1}^{d}{x_{i}^{k}},k\geq 1}$ is Schur-convex.
• ${\displaystyle \sum _{i=1}^{d}{x_{i}^{k}},0<k<1}$ is Schur-concave.
• The function ${\displaystyle f(x)=\prod _{i=1}^{d}x_{i}}$ is Schur-concave, when we assume all ${\displaystyle x_{i}>0}$. In the same way, all the elementary symmetric functions are Schur-concave, when ${\displaystyle x_{i}>0}$.
• A natural interpretation of majorization is that if ${\displaystyle x\succ y}$ then ${\displaystyle x}$ is less spread out than ${\displaystyle y}$. 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 ${\displaystyle X_{1},\dots ,X_{n}}$ are exchangeable random variables, then the function ${\displaystyle {\text{E}}\prod _{j=1}^{n}X_{j}^{a_{j}}}$ is Schur-convex as a function of ${\displaystyle a=(a_{1},\dots ,a_{n})}$, assuming that the expectations exist.
• The Gini coefficient is strictly Schur convex.

References

See also

• Quasiconvex function

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