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.
