Schur-convex function


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 is majorized by, one has that. 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.

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
for all
holds for all 1≤i≠j≤d.

Examples