Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29
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FALSE: Schwarz's lemma remains true without the hypothesis f(0)=0

Statement

Every holomorphic self-map of the unit disc satisfies the conclusions of Schwarz's lemma even without the hypothesis f(0)=0.

Facts & Assumptions

Given: The Blaschke factor f(z)=φ1/2(z).

[F1]

Every Blaschke factor is an automorphism of the unit disc, hence a holomorphic self-map of D (Blaschke factors are automorphisms of the disc).

Refutation

1.1

By [F1], f(z)=φ1/2(z) is a holomorphic self-map of D.

F1given
2.1

But f(0)=1/2, so ∣f(0)∣=1/2>0=∣0∣; this already violates the usual Schwarz-lemma bound ∣f(z)∣≤∣z∣ at z=0. Hence the fixed-point hypothesis at 0 cannot be removed.

step 1.1algebra∎

Depends on

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Sources