LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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.

Locally bounded holomorphic families are locally equicontinuous

Statement

Every locally bounded family of holomorphic functions on a plane domain is locally equicontinuous.

Facts & Assumptions

Given: A locally bounded family $F$ of holomorphic functions on a plane domain $Ω$ .

[L1]

Cauchy estimates bound derivatives on a smaller concentric disc from a common bound on a larger one (Cauchy estimates on a smaller concentric disc).

Proof

technique · direct

1.1

Fix $a \in Ω$ . Local boundedness gives a closed disc $\overline{D} (a, R) \subseteq Ω$ and a common bound $∣ f ∣ \leq M$ there for every $f \in F$ . Applying [L1] to the inner disc $\overline{D} (a, R / 2)$ gives the uniform derivative bound $∣ f^{'} (z) ∣ \leq 4 M / R$ on that smaller disc.

L1given

2.1

The smaller closed disc is convex, so integrating $f^{'}$ along the line segment from $w$ to $z$ yields $∣ f (z) - f (w) ∣ \leq (4 M / R) ∣ z - w ∣$ for every $f \in F$ . This is the local equicontinuity estimate.

givenalgebra∎

Depends on

Used by

FALSE: Arzelà-Ascoli alone proves Montel's theorem False statement
Montel's theorem: every locally bounded holomorphic family is normal Theorem

Dependency tree · two levels

14 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, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)