CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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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.

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Maximum principle for the real part of a holomorphic function

Statement

If the real part of a holomorphic function on a complex domain has an interior local maximum, then the function is constant.

That is, if $f$ is holomorphic on a complex domain $Ω$ and $Re f (z) \leq Re f (a)$ throughout some neighbourhood of $a \in Ω$ , then $f$ is constant on $Ω$ .

Facts & Assumptions

Given: A holomorphic function $f$ on a complex domain $Ω$ , a point $a \in Ω$ , and a neighbourhood $V$ on which $Re f (z) \leq Re f (a)$ .

[L1]

Every nonconstant holomorphic function on a complex domain is an open map (Open mapping theorem for holomorphic functions).

Proof

technique · direct

1.1L1given

Suppose $f$ is nonconstant. For a disc $D$ about $a$ contained in $V$ , [L1] makes $f [D]$ an open neighbourhood of $f (a)$ , so $D (f (a), ρ) \subseteq f [D]$ for some $ρ > 0$ .

2.1step 1.1algebra

The point $f (a) + ρ / 2$ belongs to that target disc and has real part $Re f (a) + ρ / 2 > Re f (a)$ , so some point of $D$ violates the assumed local maximum.

3.1step 1.1step 2.1∎

The contradiction in step 2.1 rules out nonconstancy, so $f$ is constant on $Ω$ .

Depends on

Open mapping theorem for holomorphic functions

Used by

Dependency tree · two levels

6 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, Exercise 1.16(2) (standard reference, not scraped)