Schur-convex function

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

* $sum\_\{i=1\}^d\{x\_i^k\},k\; ge\; 1$ is Schur-convex

* $prod\_\{i=1\}^d\{x\_i\}$ is Schur-concave