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In mathematics, a '''Schur-convex function''', also known as '''S-convex''', '''isotonic function''' and '''order-preserving function''' is a [[function (mathematics)|function]] <math>f: \mathbb{R}^d\rightarrow \mathbb{R}</math>, for which if <math>\forall x,y\in \mathbb{R}^d </math> where <math>x</math> is [[majorization|majorized]] by <math>y</math>, then <math>f(x)\le f(y)</math>. Named after [[Issai Schur]], Schur-convex functions are used in the study of [[majorization]]. Every function that is [[Convex function|convex]] and [[Symmetric function|symmetric]] is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).
== Schur-concave function ==
A function <math>f</math> is 'Schur-concave' if its negative,<math>-f</math>, is Schur-convex.
==A simple criterion==
If $f$ is Schur-convex and all first partial derivatives exist, then the following holds, where $f_{(i)}(x)$ denotes the partial derivative with respect to $x_i$:
$$ (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \ge 0
$$ for all $x$. Since $f$ is a symmetric function., the above condition implies all the similar conditions for the remaining indexes!
== Examples ==
* $f(x)=\min(x)$ is Schur-concave while $f(x)=\max(x)$ is Schur-convex. This can be seen directly from the definition.
* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave▼
▲* The [[Shannon entropy]] function <math>\sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}}</math> is Schur-concave.
* [[Rényi entropy]] funtion is also Schur-concave.
* <math>\sum_{i=1}^d{x_i^k},k \ge 1</math> is Schur-convex.▼
* The function $f(x) = \prod_{i=1}^n x_i is Schur-concave, when we assume all $x_i > 0$. In the same way, all the
▲* <math>\sum_{i=1}^d{x_i^k},k \ge 1</math> is Schur-convex
[[Elementary symmetric polynomial|Elementary symmetric function]]s are Schur-convex, when $x_i > 0$.
[[Category:Convex analysis]]
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