Schur-convex function explained

f: R ^{d →} R that for all x,y\in 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.

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.^[1]

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 \sum ^{n} _{i =1} g(x _{i)} 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)\left( (\partial f)/(\partial x _{i})} - (\partial f)/(\partial x _{j}) \right) \ge 0 for all x \in R ^{d}

holds for all .^[2]

Examples

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

\sum ^{d{P} _{i} ⋅ log _{2 { (1)/(P _{i})} }} is Schur-concave.

• The Rényi entropy function is also Schur-concave.

x \mapsto \sum ^{k},k} _{i} \ge 1 is Schur-convex if k \geq 1 , and Schur-concave if k \in ( 0 , 1 ) .

• The function

f(x) = \prod ^{d} _{i =1} 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 \succ y then x is less 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.

• A probability example: If

X _{1,} ... , X _{n} are exchangeable random variables, then the function E \prod ^{n} _{j =1} X ^{a _{j}} _{j} is Schur-convex as a function of a = (a _{1,} ... , a _{n)} , assuming that the expectations exist.

• The Gini coefficient is strictly Schur convex.

See also

• Quasiconvex function

Notes and References

1. Book: Roberts . A. Wayne . Convex functions . Varberg . Dale E. . 1973 . Academic Press . 9780080873725 . New York . 258 . registration.
2. Book: E. Peajcariaac. Josip. L. Tong. Y.. Convex Functions, Partial Orderings, and Statistical Applications. 3 June 1992 . Academic Press. 9780080925226. 333.