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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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

1.1

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 $ρ > 0$ .

L1given

2.1

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 $∣ f (a) ∣$ .

step 1.1algebra

3.1

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

step 1.1step 2.1∎

Depends on

Used by

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)