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

See also


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