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.
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.
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.
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}
.
-
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.
-
The Shannon entropy function
∑ i = 1 d P i ⋅ log 2 1 P i{\displaystyle \sum {i=1}^{d}{P{i}\cdot \log {2}{\frac {1}{P{i}}}}}
is Schur-concave.
-
The Rényi entropy function is also Schur-concave.
-
x ↦
∑ i = 1 d x i k , k ≥ 1{\displaystyle x\mapsto \sum {i=1}^{d}{x{i}^{k}},k\geq 1}
is Schur-convex if
k
≥
1
{\displaystyle k\geq 1}
, and Schur-concave if
k
∈
(
0
,
1
)
{\displaystyle k\in (0,1)}
.
-
The function
f ( x ) = ∏ i = 1 d x i{\displaystyle f(x)=\prod {i=1}^{d}x{i}}
is Schur-concave, when we assume all
x
i
>
0
{\displaystyle x_{i}>0}
. In the same way, all the elementary symmetric functions are Schur-concave, when
x
i
>
0
{\displaystyle x_{i}>0}
.
-
A natural interpretation of majorization is that if
x ≻ y{\displaystyle x\succ y}
then
x
{\displaystyle x}
is less spread out than
y
{\displaystyle 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{\displaystyle X_{1},\dots ,X_{n}}
are exchangeable random variables, then the function
E
∏
j
=
1
n
X
j
a
j
{\displaystyle {\text{E}}\prod _{j=1}^{n}X_{j}^{a_{j}}}
is Schur-convex as a function of
a
=
(
a
1
,
…
,
a
n
)
{\displaystyle a=(a_{1},\dots ,a_{n})}
, assuming that the expectations exist.
- The Gini coefficient is strictly Schur convex.
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- Quasiconvex function
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