The Schwarz Lemma is a statement about the growth behavior of holomorphic functions on the unit disk.
Let
D
:=
{
z
∈
C
:
|
z
|
<
1
}
{\displaystyle \mathbb {D} :=\{z\in \mathbb {C} :|z|<1\}}
be the unit disk, and let
f
:
D
→
D
{\displaystyle f\colon \mathbb {D} \to \mathbb {D} }
be holomorphic with
f
(
0
)
=
0
{\displaystyle f(0)=0}
. Then the following hold:
|
f
(
z
)
|
≤
|
z
|
{\displaystyle |f(z)|\leq |z|}
for all
z
∈
D
{\displaystyle z\in \mathbb {D} }
|
f
′
(
0
)
|
≤
1
{\displaystyle |f'(0)|\leq 1}
If
|
f
′
(
0
)
|
=
1
{\displaystyle |f'(0)|=1}
or
|
f
(
z
0
)
|
=
|
z
0
|
{\displaystyle |f(z_{0})|=|z_{0}|}
for some
z
0
∈
D
∖
0
{\displaystyle z_{0}\in \mathbb {D} \setminus {0}}
, then
f
{\displaystyle f}
is a rotation, i.e., there exists a
λ
{\displaystyle \lambda }
with
|
λ
|
=
1
{\displaystyle |\lambda |=1}
such that
f
(
z
)
=
λ
z
{\displaystyle f(z)={\lambda }z}
for all
z
∈
D
{\displaystyle z\in \mathbb {D} }
.
Define
g
:
D
→
C
{\displaystyle g\colon \mathbb {D} \to \mathbb {C} }
by
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.}
Then
g
{\displaystyle g}
is continuous and therefore, by the Riemann Removability Theorem , also holomorphic . Let
r
<
1
{\displaystyle r<1}
. By the Maximum Principle , for
|
z
|
≤
r
{\displaystyle |z|\leq r}
, we have:
|
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}}.}
As
r
→
1
{\displaystyle r\to 1}
, it follows that
|
g
(
z
)
|
≤
1
{\displaystyle |g(z)|\leq 1}
, hence
|
f
(
z
)
|
≤
|
z
|
{\displaystyle |f(z)|\leq |z|}
for all
z
∈
D
{\displaystyle z\in \mathbb {D} }
, proving the first two statements. If equality holds in either case, then
|
g
|
{\displaystyle |g|}
has a local maximum in the interior of
D
{\displaystyle \mathbb {D} }
. By the Maximum Modulus Principle,
g
{\displaystyle g}
must be constant. This constant
λ
{\displaystyle \lambda }
has modulus
1
{\displaystyle 1}
, and the claim follows.
See Fischer, p. 286.
This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Lemma_von_Schwarz