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 is also Schur-convex.

Schur-concave function

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

Examples

• Entropy function ${\displaystyle \sum _{i=1}^{d}{P_{i}\cdot \log _{2}{\frac {1}{P_{i}}}}}$  is Schur-concave

• ${\displaystyle \sum _{i=1}^{d}{x_{i}^{k}},k\geq 1}$  is Schur-convex

• ${\displaystyle \prod _{i=1}^{d}{x_{i}}}$  is Schur-concave