CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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A bounded holomorphic function on all of $C^{m}$ is constant

Statement

Let $m \geq 1$ and let $f : C^{m} \to C$ be holomorphic. If there is a real $M \geq 0$ such that $∣ f (z) ∣ \leq M$ for every $z \in C^{m}$ , then $f$ is constant.

Facts & Assumptions

Given: A holomorphic function $f : C^{m} \to C$ and a real $M \geq 0$ such that $∣ f (z) ∣ \leq M$ for every $z \in C^{m}$ .

[L1]

The composite of holomorphic maps is holomorphic and its complex Jacobian is the product (The composite of holomorphic maps is holomorphic and its complex Jacobian is the product).

[L2]

Every bounded entire function of one complex variable is constant (Liouville's theorem: every bounded entire function is constant).

[L3]

Holomorphic functions on open subsets of $C^{m}$ are those of Holomorphic functions on an open subset of $C^{m}$ , and they are continuous (A holomorphic function of several variables is continuous and separately holomorphic).

Proof

technique · direct

1.1

Fix $z \in C^{m}$ . The map $ℓ_{z} : C \to C^{m}$ defined by $ℓ_{z} (ξ) = ξ z$ is holomorphic, so by [L1] the composite $g_{z} : = f \circ ℓ_{z} : C \to C$ is holomorphic; and for every $ξ \in C$ one has $∣ g_{z} (ξ) ∣ = ∣ f (ξ z) ∣ \leq M$ , so [L2] makes $g_{z}$ constant on $C$ .

givenL1L2L3

2.1

Evaluating the constant function $g_{z}$ at $ξ = 0$ and $ξ = 1$ gives $f (z) = g_{z} (1) = g_{z} (0) = f (0)$ . Since $z$ was arbitrary, $f$ is constant on $C^{m}$ .

step 1.1∎

Remarks

No connectedness argument is needed. Every point is compared directly with the origin along the complex line it spans, so the conclusion is pointwise and not topological.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

40 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, Tasty Bits of Several Complex Variables, v4.4, Ex. 1.2.13 (standard reference, not scraped)

A bounded holomorphic function on all of CmCm is constant