Das Lemma von Schwarz ist eine Aussage über das Wachstumsverhalten holomorpher Funktionen auf der Einheitskreisscheibe.
Es sei
D
:=
{
z
∈
C
:
|
z
|
<
1
}
{\displaystyle \mathbb {D} :=\{z\in \mathbb {C} :|z|<1\}}
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}
Gilt
|
f
′
(
0
)
|
=
1
{\displaystyle |f'(0)|=1}
|
f
(
z
0
)
|
=
|
z
0
|
{\displaystyle |f(z_{0})|=|z_{0}|}
z
0
∈
D
∖
{
0
}
{\displaystyle z_{0}\in \mathbb {D} \setminus \{0\}}
f
{\displaystyle f}
λ
{\displaystyle \lambda }
|
λ
|
=
1
{\displaystyle |\lambda |=1}
f
(
z
)
=
λ
⋅
z
{\displaystyle f(z)={\lambda }\cdot z}
z
∈
D
{\displaystyle z\in \mathbb {D} }
Definiere
g
:
D
→
C
{\displaystyle g\colon \mathbb {D} \to \mathbb {C} }
g
(
z
)
=
{
f
(
z
)
z
,
z
≠
0
f
′
(
0
)
,
z
=
0
{\displaystyle g(z)=\left\{{\begin{array}{rr}\displaystyle {\frac {f(z)}{z}},&z\neq 0\\f'(0),&z=0\end{array}}\right.}
Dann ist
g
{\displaystyle g}
Riemannschen Hebbarkeitssatz auch holomorph . Sei
r
<
1
{\displaystyle r<1}
Maximumprinzip also für
|
z
|
≤
r
{\displaystyle |z|\leq r}
|
g
(
z
)
|
≤
max
|
z
|
=
r
|
g
(
z
)
|
=
max
|
z
|
=
r
|
f
(
z
)
|
r
≤
1
r
.
{\displaystyle |g(z)|\leq \max _{|z|=r}|g(z)|={\frac {\max _{|z|=r}|f(z)|}{r}}\leq {\frac {1}{r}}.}
Für
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} }
Gilt in einem der Beiden Fälle Gleichheit, so hat also
|
g
|
{\displaystyle |g|}
D
{\displaystyle \mathbb {D} }
g
{\displaystyle g}
λ
{\displaystyle \lambda }
1
{\displaystyle 1}
Vgl. Fischer S. 286.
