CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The normalized Riemann map is unique

Statement

Let $Ω ⊊ C$ be homologically simply connected and let $z_{0} \in Ω$ . If $f, g : Ω \to D$ are biholomorphic and satisfy

$f (z_{0}) = g (z_{0}) = 0, f^{'} (z_{0}) > 0, g^{'} (z_{0}) > 0,$

then $f = g$ .

Facts & Assumptions

Given: Two normalized biholomorphisms $f, g : Ω \to D$ as in the statement.

[L1]

Such maps exist by the Riemann mapping theorem (Every proper homologically simply connected plane domain is conformally equivalent to the unit disc).

[L2]

A holomorphic self-map of $D$ fixing $0$ is a rotation, and equality in Schwarz's lemma is exactly the rotational case (Schwarz lemma with the equality cases).

[L3]

Complex derivatives satisfy the chain rule (The chain rule for complex derivatives).

Proof

technique · direct

1.1

The composite $h : = g \circ f^{- 1}$ is a biholomorphic self-map of $D$ , and $h (0) = 0$ because both maps send $z_{0}$ to $0$ . Thus [L2] gives $h (ζ) = e^{i θ} ζ$ for some real $θ$ .

L1L2given

2.1

Differentiate the identity $g = h \circ f$ at $z_{0}$ . By [L3], $g^{'} (z_{0}) = h^{'} (0) f^{'} (z_{0}) = e^{i θ} f^{'} (z_{0}) .$ Since both displayed derivatives in the statement are positive real numbers, $e^{i θ} = 1$ .

L3step 1.1algebra

3.1

Hence $h$ is the identity on $D$ , so $g = h \circ f = f$ .

step 1.1step 2.1∎

Depends on

Used by

The unit-disc extremal problem is solved by the identity Example

Dependency tree · two levels

16 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

Matthias Weber, Complex Analysis, Theorem 5.2.6 (standard reference, not scraped)
Walter Rudin, Real and Complex Analysis, Theorem 14.9 (standard reference, not scraped)