CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

How statement and proof provenance work

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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.

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The family nz is not normal on any domain containing zero

Statement refuted

The family $f_{n} (z) = n z$ is normal on every plane domain containing $0$ .

Facts & Assumptions

Given: A plane domain $Ω$ containing $0$ and the functions $f_{n} (z) = n z$ .

[L1]

Normal holomorphic families are locally bounded (Normal holomorphic families are locally bounded).

Counterexample

technique · direct

1.1

Choosing a closed disc $\overline{D} (0, r) \subseteq Ω$ , the point $r / 2$ satisfies $∣ f_{n} (r / 2) ∣ = n r / 2 \to \infty$ . So the family is not locally bounded near the zero point.

givenchoosealgebra

2.1

Fact [L1] makes local boundedness necessary for normality, so this family is not normal on any domain containing $0$ .

L1given∎

Depends on

Normal holomorphic families are locally bounded

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)