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Blaschke factors are automorphisms of the disc

Statement

For each $a \in D$ , the Blaschke factor

$φ_{a} (z) = \frac{a - z}{1 - \overline{a} z}$

is a biholomorphic self-map of $D$ . More precisely,

$φ_{a} (D) = D and φ_{a} (φ_{a} (z)) = z (z \in D),$

so $φ_{a}$ is an automorphism of the disc.

Facts & Assumptions

Given: A point $a \in D$ .

[F1]

The unit disc is $D = {z \in C : ∣ z ∣ < 1}$ , and the Blaschke factor is $φ_{a} (z) = (a - z) / (1 - \overline{a} z)$ with denominator nonzero on $D$ (The unit disc, the upper half-plane, and Blaschke factors).

[F2]

A map between complex domains is biholomorphic exactly when it is bijective, holomorphic, and has holomorphic inverse (Biholomorphic maps between complex domains).

Proof

technique · direct

1.1F1givenalgebra

For $z \in D$ , [F1] gives $1 - ∣ φ_{a} (z) ∣^{2} = (∣ 1 - \overline{a} z ∣^{2} - ∣ a - z ∣^{2}) / ∣ 1 - \overline{a} z ∣^{2} = (1 - ∣ a ∣^{2}) (1 - ∣ z ∣^{2}) / ∣ 1 - \overline{a} z ∣^{2} > 0$ , so $∣ φ_{a} (z) ∣ < 1$ and $φ_{a} (D) \subseteq D$ .

2.1F1step 1.1algebra

A direct simplification from [F1] gives $φ_{a} (φ_{a} (z)) = (a - \frac{a - z}{1 - \overline{a} z}) / (1 - \overline{a} \frac{a - z}{1 - \overline{a} z}) = z$ for $z \in D$ , so $φ_{a}$ is its own inverse on $D$ .

3.1F1F2step 2.1∎

By [F1], $φ_{a}$ is holomorphic on $D$ ; step 2.1 makes it bijective with holomorphic inverse itself, so [F2] makes $φ_{a}$ biholomorphic on $D$ .

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §2 (standard reference, not scraped)
Jiri Lebl, Guide to Cultivating Complex Analysis, Proposition 3.5.2 (standard reference, not scraped)