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} that for all x,y\in \mathbb{R}^d such that x is majorized by y, one has that 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).[1]

Schur-concave function[edit]

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

Schur-Ostrowski criterion[edit]

If f is symmetric and all first partial derivatives exist, then f is Schur-convex if and only if

(x_i - x_j)(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}) \ge 0 for all x \in \mathbb{R}^d

holds for all 1≤ijd.[2]

Examples[edit]

  •  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.

References[edit]

  1. ^ Roberts, A. Wayne; Varberg, Dale E. (1973). Convex functions. New York: Academic Press. p. 258. ISBN 9780080873725. 
  2. ^ E. Peajcariaac, Josip; L. Tong, Y. Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226. 

See also[edit]