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n
1)
n
. This function is obviously symmetric on
n
. It is also increasing
defined on
I
=
(0
,
I
n
n
,
on
I
since, for all
x
∈
I
∂
∂
2
x
i
g
(
x
)
=
x
i
)
2
≥
0
,
i
=
1
,...,
n
.
x
i
−
(1
The condition (
A3.12
) becomes
x
2
)
2
x
1
(1
2
x
2
n
(
x
1
−
x
1
)
2
−
≥
0
,
∀
x
∈
I
.
x
2
)
2
−
(1
−
This can be verified by computing the derivative of
2
x
f
(
x
)
=
x
2
)
2
,
−
(1
which can be seen to be positive for all
x
∈
I
,
x
2
)(1
3
x
2
)
2(1
−
+
f
(
x
)
=
>
0
.
(1
−
x
2
)
4
Thus,
g
is Schur-convex.
A3.2.1
Specialized tests
It is possible to explicitly determine the form of condition (
A3.12
) for special cases. The
following list contains but a small sample of these results.
1. If
I
is an interval and
h
:
I
→
IR is convex, then
n
g
(
x
)
=
h
(
x
i
)
(A3.13)
i
=
1
n
.
2. Let
h
be a continuous nonnegative function on
is Schur-convex on
I
I
⊂
IR. The function
n
g
(
x
)
=
h
(
x
i
)
(A3.14)
i
=
1
n
.
3. A more general version, which subsumes the first two results, considers the compo-
sition of functions of the form
is Schur-convex on
I
if and only if log
h
(
x
) is convex on
I
=
,
,...,
,
g
(
x
)
f
(
h
(
x
1
)
h
(
x
2
)
h
(
x
n
))
(A3.15)
where
f
:IR
n
IR . I f
f
is increasing and Schur-convex and
h
is con-
vex (increasing and convex), then
g
is Schur-convex (increasing and Schur-convex).
Alternatively, if
f
is decreasing and Schur-convex and
h
is concave (decreasing and
concave), then
g
is also Schur-convex (increasing and Schur-convex).
→
IR and
h
:IR
→
These results establish that there is a connection between convexity (in the con-
ventional sense) and Schur-convexity. In addition to the cases listed above, if
g
is
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