ESP game

In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function $f:{\mathbb{ℝ}}^d\to \mathbb{ℝ}$, for which if $\mathrm{\forall }x,y\in {\mathbb{ℝ}}^d$ where $x$ is majorized by $y$, then $f(x)\le f(y)$. 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).

Schur-concave function

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

A simple criterion

If $f$ is Schur-convex and all first partial derivatives exist, then the following holds, where $f_{(i)}(x)$ denotes the partial derivative with respect to $x_i$:

$(x_1-x_2)(f_{(1)}(x)-f_{(2)}(x))\ge 0$ for all $x$. Since $f$ is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!

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.

• Some probability examples: 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.

See also

• majorization
• Quasiconvex function
• Convex function

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