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.

Schur-concave function

A function ${\displaystyle f}$  is Schur-concave if ${\displaystyle -f}$  is Schur-convex.

Examples

• Entropy