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

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Minimum modulus principle for a nowhere-zero holomorphic function

Statement

A nowhere-zero holomorphic function on a complex domain cannot have an interior local modulus minimum unless it is constant.

Equivalently, if $f$ is holomorphic and nowhere zero on a complex domain $Ω$ and $∣ f (z) ∣ \geq ∣ f (a) ∣$ on some neighbourhood of $a \in Ω$ , then $f$ is constant.

Facts & Assumptions

Given: A nowhere-zero holomorphic function $f$ on a complex domain $Ω$ and an interior local minimum of $∣ f ∣$ at $a$ . The modulus is multiplicative, so $∣ 1 / f ∣ = 1 / ∣ f ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[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).

[L2]

A nowhere-zero holomorphic function has a holomorphic reciprocal (Linearity, product, reciprocal, and quotient rules for complex derivatives).

Proof

technique · direct

1.1

Since $f$ is nowhere zero, [L2] makes $1 / f$ holomorphic throughout $Ω$ .

L2given

2.1

The local inequality $∣ f (z) ∣ \geq ∣ f (a) ∣ > 0$ is equivalent to $∣ 1 / f (z) ∣ \leq ∣ 1 / f (a) ∣$ , so $∣ 1 / f ∣$ has an interior local maximum at $a$ . By [L1], $1 / f$ is constant.

step 1.1L1algebra

3.1

The reciprocal of that nonzero constant is $f$ , so $f$ is constant on $Ω$ .

step 2.1algebra∎

Depends on

Used by

Constant boundary modulus forces an interior zero or constancy Corollary
FALSE: every interior local modulus minimum forces constancy False statement

Dependency tree · two levels

12 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

J. Lebl, Guide to Cultivating Complex Analysis, Exercise 3.3.18 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, Theorem 1.17 (standard reference, not scraped)