Schur-convex function

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

A simple criterion

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

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

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.

• The function ${\displaystyle f(x)=\prod _{i=1}^{n}x_{i}}$  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 more 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.

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

• Some probability examples: 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.

See also

• majorization
• Quasiconvex function
• Convex function