Schur-convex function

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

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

Schur-Ostrowski criterion

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

References

  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


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