TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29

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Every automorphism of the disc is a rotated Blaschke factor

Statement

A holomorphic map $f : D \to D$ is an automorphism of $D$ if and only if there exist $a \in D$ and $θ \in R$ such that

$f (z) = e^{i θ} φ_{a} (z) = e^{i θ} \frac{a - z}{1 - \overline{a} z} (z \in D) .$

Facts & Assumptions

Given: A holomorphic self-map $f : D \to D$ .

[F1]

The automorphism group $Aut (D)$ consists of the biholomorphic self-maps of $D$ (Conformal equivalence and the automorphism group of a domain).

[F2]

Every Blaschke factor is an automorphism of $D$ (Blaschke factors are automorphisms of the disc).

[F3]

Equality in Schwarz's lemma characterizes rotations (Schwarz lemma with the equality cases).

Proof

technique · direct

1.1

Assume first that $f \in Aut (D)$ , and let $a = f^{- 1} (0)$ . By [F2], the map $g : = f \circ φ_{a}$ is an automorphism of $D$ with $g (0) = 0$ .

F1F2givenconstruct

2.1

Applying [F3] to $g$ and to its inverse $g^{- 1}$ shows $∣ g (z) ∣ \leq ∣ z ∣$ and $∣ z ∣ = ∣ g^{- 1} (g (z)) ∣ \leq ∣ g (z) ∣$ for every $z \in D$ , so $∣ g (z) ∣ = ∣ z ∣$ throughout $D$ . Hence [F3] forces $g (z) = e^{i θ} z$ for some real $θ$ .

F1F3step 1.1algebra

3.1

Therefore $f (z) = g (φ_{a} (z)) = e^{i θ} φ_{a} (z)$ . Conversely, if $f (z) = e^{i θ} φ_{a} (z)$ , then the rotation $z \mapsto e^{i θ} z$ and the Blaschke factor $φ_{a}$ are automorphisms, so [F2] and [F1] make $f$ an automorphism.

F1F2step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

8 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, Theorem 2.2 (standard reference, not scraped)
Jiri Lebl, Guide to Cultivating Complex Analysis, Proposition 3.5.3 (standard reference, not scraped)