In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function that for all such that x is majorized by y, one has that. Named after Issai Schur, Schur-convex functions are used in the study of majorization. A function f is 'Schur-concave' if its negative, −f, is Schur-convex.

Properties

Every function that is convex and symmetric (under permutations of the arguments) is also Schur-convex. Every Schur-convex function is symmetric, but not necessarily convex. If f is (strictly) Schur-convex and g is (strictly) monotonically increasing, then g\circ f is (strictly) Schur-convex. If g is a convex function defined on a real interval, then 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 holds for all.

Examples