Math 411: Honours Complex Variables - University of Alberta

50CHAPTER7. ELEMENTARYPROPERTIESOFHOLOMORPHICFUNCTIONS
Definition. Let D1,D2 ⊂ C be open. Then f: D1 → D2 is called biholomorphic (or
conformal) if
(a) f is bijective and
(b) both f and f −1 are holomorphic.
Corollary 7.4.1. Let f : D → D be biholomorphic such that f(0) = 0. Then there
exists c ∈ C with |c|= 1 such that f(z) = cz for z ∈ D.
Proof. Let z ∈ D. Then |f(z)|≤ |z| holds by Schwarz’s Lemma, as does
|z|= |f −1 (f(z))|≤ |f(z)|.
The result then follows from Schwarz’s Lemma.
Lemma 7.2. Let w ∈ D, and define
Then:
(i) φw maps D bijectively onto D;
(ii) φw(w) = 0;
(iii) φw(0) = w;
(iv) φ −1
w = φw.
φw: D → C, z ↦→
w −z
1− ¯wz .
Proof. Obviously, φw is holomorphic and extends continuously to D.
Then for |z|= 1 we may express
� �
�
|φw(z)|= �
w −z �
�
1
�1−
¯wz � |¯z| =
� �
�
�
w−z �
�
�¯z
− ¯w � = 1.
By the Maximum Modulus Principle, φw(D) ⊂ D holds. Since φw is not constant,
φw(D) is open and thus contained in the interior of D, i.e. in D.
It is obvious that φw(w) = 0 and φw(0) = w.
Moreover, we have for z ∈ D:
(φw ◦φw)(z) =
Hence, φw is bijective with φ −1
w = φw.
= w(1− ¯wz)−(w −z)
w− w−z
1−¯wz
1− ¯w w−z
1−¯wz
(1− ¯wz)− ¯w(w −z)
= −|w|2 z +z
1−|w| 2
= z.