LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cauchy estimates propagate from a compact set to its hull

Statement

Let $Ω \subseteq C^{m}$ be a domain, let $\emptyset \neq K ⋐ Ω$ be compact, let $0 < r < δ_{Ω} (K)$ , and define

$K_{r} : = ⋃_{a \in K} {\overline{Δ}}_{r} (a) .$

Then $K_{r} ⋐ Ω$ . Moreover, for every holomorphic $f$ on $Ω$ and every multi-index $α$ ,

$\sup_{w \in {\hat{K}}_{Ω}} ∣ \partial^{α} f (w) ∣ \leq α! r^{- (α_{1} + \dots + α_{m})} \sup_{z \in K_{r}} ∣ f (z) ∣ .$

Facts & Assumptions

Given: A nonempty compact set $K ⋐ Ω$ , a number $0 < r < δ_{Ω} (K)$ , and $f \in O (Ω)$ .

[L1]

The hull ${\hat{K}}_{Ω}$ is characterized by the inequalities against holomorphic functions on $Ω$ (Holomorphic hulls and holomorphic convexity).

[L2]

If a holomorphic function is defined on a polydisc, then its mixed derivatives satisfy the several-variable Cauchy estimates there (Cauchy estimates for mixed derivatives on a polydisc).

[L3]

The boundary-radius inequality $r < δ_{Ω} (K)$ means that $Δ_{r} (a) \subseteq Ω$ for every $a \in K$ (The equal-radius polydisc boundary function).

Proof

technique · direct

1.1L3given

By [L3], every closed polydisc ${\overline{Δ}}_{r} (a)$ with $a \in K$ lies in $Ω$ . Since $K$ is compact, the union $K_{r}$ is bounded and closed in $C^{m}$ , hence compact, and it lies in $Ω$ . Thus $K_{r} ⋐ Ω$ .

1.2L2given

Fix $a \in K$ . Because ${\overline{Δ}}_{r} (a) \subseteq Ω$ , the function $f$ is holomorphic on $Δ_{r} (a)$ , so [L2] gives $∣ \partial^{α} f (a) ∣ \leq α! r^{- (α_{1} + \dots + α_{m})} \sup_{z \in {\overline{Δ}}_{r} (a)} ∣ f (z) ∣ \leq α! r^{- (α_{1} + \dots + α_{m})} \sup_{z \in K_{r}} ∣ f (z) ∣ .$ Therefore the holomorphic function $\partial^{α} f$ is bounded on $K$ by the displayed constant.

2.1L1step 1.2∎

The function $\partial^{α} f$ is holomorphic on $Ω$ , so [L1] propagates the step-1.2 bound from $K$ to ${\hat{K}}_{Ω}$ . This is exactly the stated estimate.

Depends on

Used by

Cartan-Thullen boundary-radius theorem Theorem

Dependency tree · two levels

18 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

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.5 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)