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 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)\le 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[edit]

A function f is 'Schur-concave' if its negative,-f, is Schur-convex.

A simple criterion[edit]

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!


  •  f(x)=\min(x) is Schur-concave while  f(x)=\max(x) is Schur-convex. This can be seen directly from the definition.
  •  \sum_{i=1}^d{x_i^k},k \ge 1 is Schur-convex.
  • If  g is a convex function defined on a real interval, then  \sum_{i=1}^n g(x_i) is Schur-convex.
  • A probability example: If  X_1, \dots, X_n are exchangeable random variables, then the function   \text{E} \prod_{j=1}^n X_j^{a_j} is Schur-convex as a function of  a=(a_1, \dots, a_n) , assuming that the expectations exist.

See also[edit]