False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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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.
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FALSE: the local maximum modulus principle needs no connectedness

Statement

Every holomorphic function on an open subset of $C$ whose modulus has an interior local maximum is constant on that open set, even when the open set is disconnected.

Facts & Assumptions

Given: The disjoint open discs $D (- 2, 1)$ and $D (2, 1)$ in the complex metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) and their union $U$ .

[L1]

If the modulus of a holomorphic function on a complex domain has an interior local maximum, then the function is constant (Local maximum modulus principle).

Refutation

technique · direct

1.1

The two discs are nonempty, open, and disjoint, so $U = D (- 2, 1) \cup D (2, 1)$ is open but disconnected and therefore is not a complex domain (A complex domain is a nonempty connected open subset of $C$ ).

given

2.1

Define $f = 0$ on $D (- 2, 1)$ and $f = 1$ on $D (2, 1)$ . Every point has a neighbourhood on which $f$ is constant, so $f$ is holomorphic on $U$ .

step 1.1algebra

3.1

At every point of the second disc, $∣ f ∣ = 1$ is a local maximum, but $f$ is not constant on $U$ because it is $0$ on the first disc. Thus the statement is false; the connected-domain hypothesis in [L1] is exactly what prevents this componentwise witness.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

B. V. Shabat, Introduction to Complex Analysis, Theorem 1.14 (standard reference, not scraped)