Schur-convex function

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

f

:

R

d

→

R

{\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} }

that for all

x

,

y

∈

R

d

{\displaystyle x,y\in \mathbb {R} ^{d}}

such that

x

{\displaystyle x}

is majorized by

y

{\displaystyle y}

, one has that

f

(

x

)

≤

f

(

y

)

{\displaystyle f(x)\leq 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.

If

f

{\displaystyle f}

is (strictly) Schur-convex and

g

{\displaystyle g}

is (strictly) monotonically increasing, then

g

∘

f

{\displaystyle g\circ f}

is (strictly) Schur-convex.

If

g

{\displaystyle g}

is a convex function defined on a real interval, then

∑

i

=

1

n

g

(

x

i

)

{\displaystyle \sum _{i=1}^{n}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

)

(

∂

f

∂

x

i

−

∂

f

∂

x

j

)

≥

0

{\displaystyle (x_{i}-x_{j})\left({\frac {\partial f}{\partial x_{i}}}-{\frac {\partial f}{\partial x_{j}}}\right)\geq 0}

for all

x

∈

R

d

{\displaystyle x\in \mathbb {R} ^{d}}

holds for all

1

≤

i

,

j

≤

d

{\displaystyle 1\leq i,j\leq d}

.

Examples

f

(

x

)

=

min

(

x

)

{\displaystyle f(x)=\min(x)}

is Schur-concave while

f

(

x

)

=

max

(

x

)

{\displaystyle f(x)=\max(x)}

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