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

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Local maximum modulus principle

Statement

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

Precisely, if $f$ is holomorphic on a complex domain $Ω$ and there are $a \in Ω$ and a neighbourhood $V \subseteq Ω$ of $a$ such that $∣ f (z) ∣ \leq ∣ f (a) ∣$ for every $z \in V$ , 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 $∣ f (z) ∣ \leq ∣ f (a) ∣$ . The modulus obeys the usual multiplicative and positivity laws (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[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. Choose an open disc $D$ about $a$ contained in $V$ . By [L1], $f [D]$ is an open set containing $f (a)$ , so it contains a disc $D (f (a), ρ)$ for some $ρ > 0$ .

2.1step 1.1algebra

If $f (a) \neq 0$ , the point $w = (1 + t) f (a)$ lies in $D (f (a), ρ)$ and has $∣ w ∣ > ∣ f (a) ∣$ for sufficiently small $t > 0$ . If $f (a) = 0$ , any nonzero $w$ with $∣ w ∣ < ρ$ has larger modulus. Thus in either case $f [D]$ contains a value whose modulus is greater than $∣ f (a) ∣$ .

3.1step 1.1step 2.1∎

Step 2.1 contradicts the local maximum on $V$ . Hence $f$ cannot be nonconstant and must be constant on $Ω$ .

Depends on

Used by

Inner, singular inner and outer functions Definition
Finite Blaschke products Example
FALSE: the local maximum modulus principle needs no connectedness False statement
Properties of outer functions Lemma
An interior local maximum of the modulus forces a scalar holomorphic function to be constant Theorem
Boundary maximum modulus principle on a bounded domain Theorem
Hadamard factorization for finite-order entire functions Theorem
Inner-outer factorisation of a Hardy-space function Theorem
Maximum modulus principle with boundary and infinity control Theorem
Minimum modulus principle for a nowhere-zero holomorphic function Theorem
Properties of the singular functions S_μ Theorem
Schwarz lemma with the equality cases Theorem
Zero-free inner functions are unimodular multiples of singular inner functions Theorem

Dependency tree · two levels

13 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, Theorem 3.3.6 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, Theorem 1.14 (standard reference, not scraped)
J. A. Tropp, Matrix Analysis, Proposition 7.11 (standard reference, not scraped)