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).
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!
- is Schur-concave while is Schur-convex. This can be seen directly from the definition.
- The Shannon entropy function is Schur-concave.
- The Rényi entropy function is also Schur-concave.
- is Schur-convex.
- 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.
- If is a convex function defined on a real interval, then is Schur-convex.
- 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.
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