In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.
Let
D=\{z:|z|<1\}
C
f:D → C
f(0)=0
|f(z)|\leq1
D
Then
|f(z)|\leq|z|
z\inD
|f'(0)|\leq1
Moreover, if
|f(z)|=|z|
z
|f'(0)|=1
f(z)=az
a\inC
|a|=1
The proof is a straightforward application of the maximum modulus principle on the function
g(z)=\begin{cases}
f(z) | |
z |
&ifz ≠ 0\\ f'(0)&ifz=0, \end{cases}
which is holomorphic on the whole of
D
f
Dr=\{z:|z|\ler\}
r
r<1
z\inDr
zr
Dr
|g(z)|\le|g(zr)|=
|f(zr)| | |
|zr| |
\le
1 | |
r |
.
As
r → 1
|g(z)|\leq1
Moreover, suppose that
|f(z)|=|z|
z\inD
|f'(0)|=1
|g(z)|=1
D
g(z)
a
|a|=1
f(z)=az
A variant of the Schwarz lemma, known as the Schwarz - Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of the unit disc to itself:
Let
f:D\toD
z1,z2\inD
\left| | f(z1)-f(z2) |
1-\overline{f(z1) |
f(z2)}\right|\le\left|
z1-z2 | |
1-\overline{z1 |
z2}\right|
and, for all
z\inD
\left|f'(z)\right| | |
1-\left|f(z)\right|2 |
\le
1 | |
1-\left|z\right|2 |
.
The expression
d(z1,z
-1 | ||
\left| | ||
2)=\tanh |
z1-z2 | |
1-\overline{z1 |
z2}\right|
is the distance of the points
z1
z2
f
H
Letbe holomorphic. Then, for allf:H\toH
,z1,z2\inH
\left| f(z1)-f(z2) \overline{f(z1) -f(z2)}\right|\le
\left|z1-z2\right| \left|\overline{z1 -z2\right|}.
W(z)=(z-i)/(z+i)
H
D
W\circf\circW-1
D
D
W
z\inH
\left|f'(z)\right| | |
Im(f(z)) |
\le
1 | |
Im(z) |
.
If equality holds for either the one or the other expressions, then
f
f(z)= | az+b |
cz+d |
with
a,b,c,d\inR
ad-bc>0
The proof of the Schwarz - Pick theorem follows from Schwarz's lemma and the fact that a Möbius transformation of the form
z-z0 | |
\overline{z0 |
z-1}, |z0|<1,
maps the unit circle to itself. Fix
z1
M(z)= | z1-z |
1-\overline{z1 |
z}, \varphi(z)=
f(z1)-z | |
1-\overline{f(z1) |
z}.
Since
M(z1)=0
\varphi(f(M-1(z)))
0
0
\left|\varphi\left(f(M-1(z))\right)\right|=\left|
| ||||||||||
1-\overline{f(z1) |
f(M-1(z))}\right|\le|z|.
Now calling
-1 | |
z | |
2=M |
(z)
\left| | f(z1)-f(z2) |
1-\overline{f(z1) |
f(z2)}\right|\le\left|
z1-z2 | |
1-\overline{z1 |
z2}\right|.
To prove the second part of the theorem, we rearrange the left-hand side into the difference quotient and let
z2
z1
The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds.
De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of
f
0
f
The Koebe 1/4 theorem provides a related estimate in the case that
f