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 \( f(x)\le f(y) \). 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 its negative,-f, is Schur-convex.

Examples

The Shannon 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

Mathematics Encyclopedia