Schur-convex function

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ that for all $x,y\in \mathbb{R}^d$ 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. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).^[1]

Schur-concave function

A function f is 'Schur-concave' if its negative, -f, 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

$(x_i - x_j)(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}) \ge 0$ for all $x \in \mathbb{R}^d$

holds for all 1≤i≠j≤d.^[2]

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.
• $\sum_{i=1}^d{x_i^k},k \ge 1$ is Schur-convex.
• The function $f(x) = \prod_{i=1}^n x_i$ 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 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.
• If $g$ is a convex function defined on a real interval, then $\sum_{i=1}^n g(x_i)$ is Schur-convex.
• A probability example: If $X_1, \dots, X_n$ are exchangeable random variables, then the function $\text{E} \prod_{j=1}^n X_j^{a_j}$ is Schur-convex as a function of $a=(a_1, \dots, a_n)$, assuming that the expectations exist.
• The Gini coefficient is strictly Schur concave.

References

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

See also

• Quasiconvex function