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. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).

Schur-concave function

A function $f$ is 'Schur-concave' if its negative,$-f$, 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 $\sum _{i=1}^{d}{P_{i}\cdot \log _{2}{\frac {1}{P_{i}}}}$ is Schur-concave.

• Rényi entropy funtion is also Schur-concave.

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

  Elementary symmetric functions are Schur-convex, when $x_i > 0$.