Schur-convex function

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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.

[edit] Schur-concave function

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

[edit] 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
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