# 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!