3.5 The Group of Holomorphic Automorphisms on the Unit Disk
The following important result can be directly obtained from the Maximum Modulus Principle.
Lemma 3.5.1 (Schwarz).
If is holomorphic and , then
Any one of the inequalities becomes equalities iff is in the form of , where . In other words, is a pure rotation.
Proof.
Define the auxiliary function
Because , is holomorphic on . Since is an automorphism on the open disk, , . By the Maximum Modulus Principle (Theorem 3.4.1), , ,
As , we obtain that , , or that . Let . Then we get .
For the sake of the equality condition, assume . Then on the unit open disk. By Theorem 3.4.1, where and on .
Next, assume only that . It follows that . Since for all , it follows from Theorem 3.4.1 that is constant with magnitude , or in the form of , where is a constant. Consequently, .
To discuss the main topic of this section, we will first introduce the concept of a group.
Definition 3.5.1 (Group).
A group is a nonempty set and a binary operation (we will denote this as ) satisfying the four group axioms:
- Closure: , .
- Associativity: , .
- Identity Element: such that , . Note that is unique; if were both identity elements, then , and are equal.
- Inverse Element: , such that , where is the identity element. Note that is unique. Assume were both inverses of . Then, , and are equal.
A subgroup of is a subset of that is also a group under the same operation as . This relationship is denoted by or for proper subgroups.
Group operations are not necessarily commutative. In the case that they are, (specifically if ), then is an abelian group.
If is connected and is holomorphic on and bijective, is a holomorphic automorphism on . The group of holomorphic automorphisms on is denoted by , which is the set of all holomorphic automorphisms such as , with the operation of composition .
First we will show that ,
Firstly, the function is holomorphic on as , , the denominator never vanishes. Additionally, .
First, we will observe the image of . Let . Then,
Therefore, the image of lies on , and since is holomorphic and non-constant, by the Maximum Modulus Principle (Theorem 3.4.1), for any , . Therefore, maps to . We next aim to show that is bijective.
Let us first confirm injectivity. For all , we will observe when
is satisfied. It follows that
Then,
Since , then , and we get . This proves the univalence of .
Next, we will solve for the inverse of . Let . Then,
It follows that . Thus is surjective and a bijective automorphism. It follows that (3.5.1) is true. Functions in the form of (where ) are known as Möbius transformations, and the group of all such transformations is known as the Möbius transformation group on the unit disk, which is a subgroup of . Functions in the form of , where is constant, form a group known as the rotation group, which is also a subgroup of .
Theorem 3.5.1 (The Holomorphic Automorphism Group on ).
, is a composition between a Möbius transformation and a rotation. In other words, and such that
Moreover, all such functions are in .
Proof.
Define the auxiliary function
It follows that . Furthermore,
By the Schwarz Lemma (Lemma 3.5.1), . Since with , . Then,
Then, , and by the equality statement of Lemma 3.5.1,
for some constant , and
By (3.5.2), it follows that
As a direct consequence of Theorem 3.5.1, we have the following result:
Lemma 3.5.2 (Schwarz–Pick).
Let be holomorphic. For all , let and . Then,
and
The equalities hold iff .
Proof.
Let
It follows that
Then by the Schwarz Lemma (Lemma 3.5.1), for ,
Let . Then,
confirming (3.5.3). By the second statement of the Schwarz Lemma (Lemma 3.5.1), .
By the chain rule,
Let us now calculate the derivatives of and . By the quotient rule,
and
Since both derivatives are positive,
Since is arbitrary, it follows that
By the Schwarz Lemma (Lemma 3.5.1), under the equality condition that
we have that
where is constant. It follows that
Remark.
In Section 8, we will introduce the hyperbolic metric on , defined as
From (3.5.5), we get that the hyperbolic metric does in.notcrease under a holomorphic mapping of to itself. This metric is invariant (the equality condition) under all functions in . This gives a geometric explanation for Lemma 3.5.1.