In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f : R d → R that for all x , y ∈ R d such that x is majorized by y , one has that f ( x ) ≤ 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).
A function f is 'Schur-concave' if its negative, -f, is Schur-convex.
If f is symmetric and all first partial derivatives exist, then f is Schur-convex if and only if
( x i − x j ) ( ∂ f ∂ x i − ∂ f ∂ x j ) ≥ 0 for all x ∈ R d
holds for all 1≤i≠j≤d.
f ( x ) = min ( x ) is Schur-concave while f ( x ) = max ( x ) is Schur-convex. This can be seen directly from the definition.The Shannon entropy function ∑ i = 1 d P i ⋅ log 2 1 P i is Schur-concave.The Rényi entropy function is also Schur-concave. ∑ i = 1 d x i k , k ≥ 1 is Schur-convex.The function f ( x ) = ∏ i = 1 n x i is Schur-concave, when we assume all x i > 0 . In the same way, all the Elementary symmetric functions are Schur-concave, when x i > 0 .A natural interpretation of majorization is that if x ≻ y then x is more spread out than y . So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the Median absolute deviation is not.If g is a convex function defined on a real interval, then ∑ i = 1 n g ( x i ) is Schur-convex.A probability example: If X 1 , … , X n are exchangeable random variables, then the function E ∏ j = 1 n X j a j is Schur-convex as a function of a = ( a 1 , … , a n ) , assuming that the expectations exist.The Gini coefficient is strictly Schur concave.