This article is about a complex calculus lemma. It should not be confused with
Black's phrase .
The Schwarz Lemma (by Hermann Schwarz ) is a set of function theory about holomorphic self-images of the unit disk, which detects the zero point blank.
statement
Let it denote the open unit disk. Be a holomorphic function with . Then applies to all and . If in one point the equality also exists or applies, then a rotation, i.e. H. for a suitable .
D.
: =
{
z
∈
C.
:
|
z
|
<
1
}
{\ displaystyle \ mathbb {D}: = \ left \ {z \ in \ mathbb {C} \,: \, | z | <1 \ right \}}
f
:
D.
→
D.
{\ displaystyle f \ colon \ mathbb {D} \ to \ mathbb {D}}
f
(
0
)
=
0
{\ displaystyle f (0) = 0}
|
f
(
z
)
|
≤
|
z
|
{\ displaystyle | f (z) | \ leq | z |}
z
∈
D.
{\ displaystyle z \ in \ mathbb {D}}
|
f
′
(
0
)
|
≤
1
{\ displaystyle | f '(0) | \ leq 1}
z
0
∈
D.
,
z
0
≠
0
,
{\ displaystyle z_ {0} \ in \ mathbb {D}, z_ {0} \ neq 0,}
|
f
(
z
0
)
|
=
|
z
0
|
{\ displaystyle | f (z_ {0}) | = | z_ {0} |}
|
f
′
(
0
)
|
=
1
{\ displaystyle | f '(0) | = 1}
f
{\ displaystyle f}
f
(
z
)
=
e
i
λ
⋅
z
{\ displaystyle f (z) = e ^ {i \ lambda} \ cdot z}
λ
∈
R.
{\ displaystyle \ lambda \ in \ mathbb {R}}
proof
Let be the Taylor expansion of around the point . Because is so the function
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
{\ displaystyle f (z) = \ sum _ {n = 0} ^ {\ infty} a_ {n} z ^ {n}}
f
{\ displaystyle f}
z
=
0
{\ displaystyle z = 0}
f
(
0
)
=
0
{\ displaystyle f (0) = 0}
a
0
=
0
{\ displaystyle a_ {0} = 0}
G
(
z
)
: =
{
f
(
z
)
z
,
if
z
≠
0
,
f
′
(
0
)
,
otherwise
{\ displaystyle g (z): = {\ begin {cases} {\ frac {f (z)} {z}}, & {\ text {if}} z \ not = 0, \\ f '(0) , & {\ text {otherwise}} \ end {cases}}}
is holomorphic and has the Taylor expansion around zero. After the maximum principle the function takes on the circle , their maximum on the edge of. The following applies there:
D.
{\ displaystyle \ mathbb {D}}
G
(
z
)
=
∑
n
=
1
∞
a
n
z
n
-
1
{\ displaystyle g (z) = \ sum _ {n = 1} ^ {\ infty} a_ {n} z ^ {n-1}}
G
{\ displaystyle g}
K
r
: =
{
z
∈
C.
∣
|
z
|
≤
r
}
{\ displaystyle K_ {r}: = \ {z \ in \ mathbb {C} \ mid | z | \ leq r \}}
r
∈
(
0
,
1
)
{\ displaystyle r \ in (0,1)}
∂
K
r
=
{
z
∈
C.
∣
|
z
|
=
r
}
{\ displaystyle \ partial K_ {r} = \ {z \ in \ mathbb {C} \ mid | z | = r \}}
|
G
(
z
)
|
=
|
f
(
z
)
z
|
=
|
f
(
z
)
|
r
≤
1
r
,
{\ displaystyle | g (z) | = \ left | {\ frac {f (z)} {z}} \ right | = {\ frac {| f (z) |} {r}} \ leq {\ frac {1} {r}},}
so that | g (z) | is limited to all through . Since is arbitrary, the border crossing already follows and thus for everyone
. Furthermore is .
K
r
{\ displaystyle K_ {r}}
1
r
{\ displaystyle {\ frac {1} {r}}}
0
<
r
<
1
{\ displaystyle 0 <r <1}
r
→
1
{\ displaystyle r \ to 1}
|
G
(
z
)
|
≤
1
{\ displaystyle | g (z) | \ leq 1}
|
f
(
z
)
|
≤
|
z
|
{\ displaystyle | f (z) | \ leq | z |}
z
∈
D.
{\ displaystyle z \ in \ mathbb {D}}
|
f
′
(
0
)
|
=
|
G
(
0
)
|
≤
1
{\ displaystyle | f '(0) | = | g (0) | \ leq 1}
Additionally, if one with exists or is true, then there is one with . With the maximum principle it follows that is constant, i.e. for a with . So it applies .
z
0
∈
D.
{\ displaystyle z_ {0} \ in \ mathbb {D}}
|
f
(
z
0
)
|
=
|
z
0
|
{\ displaystyle | f (z_ {0}) | = | z_ {0} |}
|
f
′
(
0
)
|
=
1
{\ displaystyle | f '(0) | = 1}
z
0
∈
D.
{\ displaystyle z_ {0} \ in \ mathbb {D}}
|
G
(
z
0
)
|
=
1
{\ displaystyle | g (z_ {0}) | = 1}
G
{\ displaystyle g}
G
(
z
)
=
c
{\ displaystyle g (z) = c}
c
{\ displaystyle c}
|
c
|
=
1
{\ displaystyle | c | = 1}
f
(
z
)
=
c
⋅
z
{\ displaystyle f (z) = c \ cdot z}
Applications
Determination of holomorphic automorphism of the unit disc: .
A.
u
t
(
D.
)
=
{
f
(
z
)
=
e
i
λ
z
-
z
0
1
-
z
0
¯
z
,
λ
∈
[
0
,
2
π
)
,
z
0
∈
D.
}
{\ displaystyle \ mathrm {Aut} (\ mathbb {D}) = \ left \ {f (z) = e ^ {i \ lambda} {\ frac {z-z_ {0}} {1 - {\ overline { z_ {0}}} z}} \;, \; \ lambda \ in [0,2 \ pi), \; z_ {0} \ in \ mathbb {D} \ right \}}
From this one can determine and obtain the automorphism group of the upper half-plane .
H
{\ displaystyle \ mathbb {H}}
A.
u
t
(
H
)
≅
P
S.
L.
(
2
,
R.
)
{\ displaystyle \ mathrm {Aut} (\ mathbb {H}) \ cong PSL (2, \ mathbb {R})}
Schwarz's lemma is one of the tools used in the modern proof of Riemann's mapping theorem carried out with the help of normal families .
Lemma von Schwarz-Pick : For holomorphic functions applies to all .
f
:
D.
→
D.
{\ displaystyle f: \ mathbb {D} \ to \ mathbb {D}}
|
f
′
(
z
)
|
1
-
|
f
(
z
)
|
2
≤
1
1
-
|
z
|
2
{\ displaystyle {\ frac {| f '(z) |} {1- | f (z) | ^ {2}}} \ leq {\ frac {1} {1- | z | ^ {2}}} }
z
∈
D.
{\ displaystyle z \ in \ mathbb {D}}
Tightening
Among other things, Schwarz's lemma states that the condition applies to a holomorphic function with expansion in the power series . Ludwig Bieberbach showed that this also applies to injective functions , and put forward the Bieberbach hypothesis , later named after him , that for all . This conjecture was proven in 1985 by Louis de Branges de Bourcia .
f
:
D.
→
D.
{\ displaystyle f: \ mathbb {D} \ to \ mathbb {D}}
f
(
0
)
=
0
{\ displaystyle f (0) = 0}
f
(
z
)
=
∑
j
=
1
∞
a
j
z
j
{\ displaystyle f (z) = \ sum _ {j = 1} ^ {\ infty} a_ {j} z ^ {j}}
|
a
1
|
≤
1
{\ displaystyle | a_ {1} | \ leq 1}
|
a
2
|
≤
2
{\ displaystyle | a_ {2} | \ leq 2}
|
a
j
|
≤
j
{\ displaystyle | a_ {j} | \ leq j}
j
∈
N
{\ displaystyle j \ in \ mathbb {N} \;}
literature
Wolfgang Fischer, Ingo Lieb: Function theory. Vieweg Verlag, Braunschweig 2003, ISBN 3-528-77247-6
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