# 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 , for which if where is majorized by , then . 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 is 'Schur-concave' if its negative,, is Schur-convex.

## A simple criterion

If is Schur-convex and all first partial derivatives exist, then the following holds, where denotes the partial derivative with respect to :

- for all . Since is a symmetric function, the above condition implies all the similar conditions for the remaining indexes!

## Examples

- The Shannon entropy function is Schur-concave.

- The Rényi entropy function is also Schur-concave.

- The function is Schur-concave, when we assume all . In the same way, all the Elementary symmetric functions are Schur-concave, when .

- A natural interpretation of majorization is that if then is more spread out than . So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the Median absolute deviation is not.

- A probability example: If are exchangeable random variables, then the function is Schur-convex as a function of , assuming that the expectations exist.

- The Gini coefficient is strictly Schur concave.