False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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FALSE: boundary control alone gives the maximum principle on an unbounded domain

Statement

If a function is continuous on the closure of an unbounded complex domain, holomorphic inside, and has boundary modulus at most $1$ , then its modulus is at most $1$ throughout the domain.

Facts & Assumptions

Given: The upper half-plane $H = {z : Im z > 0}$ and $f (z) = \exp (- i z)$ . The exponential is entire and holomorphic compositions obey the complex chain rule (The complex exponential is entire and its complex derivative is itself, The chain rule for complex derivatives). The bounded-domain theorem is Boundary maximum modulus principle on a bounded domain.

[L1]

Boundary control together with control at infinity bounds the modulus throughout an unbounded complex domain (Maximum modulus principle with boundary and infinity control).

[L2]

For real $x, y$ , $∣ \exp (x + i y) ∣ = e^{x}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

Refutation

technique · direct

1.1

The function $f (z) = \exp (- i z)$ is entire and hence is holomorphic on $H$ and continuous on its closed half-plane.

L2given

2.1

For $z = x + i y$ , one has $- i z = y - i x$ , so [L2] gives $∣ f (x + i y) ∣ = e^{y}$ . Thus $∣ f (x) ∣ = 1$ on the real boundary $y = 0$ , while $∣ f (i y) ∣ = e^{y}$ is unbounded as $y \to + \infty$ .

step 1.1L2algebra

3.1

Step 2.1 violates the proposed conclusion. The valid unbounded-domain result [L1] requires control at infinity as well as finite-boundary control, and this example fails exactly that additional hypothesis.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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. A. Tropp, Matrix Analysis, Lecture 7, §7.2.2 (standard reference, not scraped)