Schur-convex function

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

Schur-concave function

A function $f$ is Schur-concave if $-f$ is Schur-convex.

Examples

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

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

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