In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function is majorized. 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.

is (strictly) Schur-convex.

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

.

Examples

• is Schur-convex. This can be seen directly from the definition.

• The Shannon entropy is Schur-concave.

• The Rényi entropy function is also Schur-concave.

• .

• . In the same way, all the elementary symmetric functions.

• A natural interpretation of majorization. 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.

• are exchangeable random variables, assuming that the expectations exist.

• The Gini coefficient is strictly Schur convex.

References

See also

• Quasiconvex function