LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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The extremal derivatives are positive and have a finite supremum

Statement

Let $Ω ⊊ C$ be homologically simply connected and let $z_{0} \in Ω$ . Then the set

$E : = {f^{'} (z_{0}) : f \in F (Ω, z_{0})}$

is a nonempty subset of $(0, \infty)$ with finite supremum.

Facts & Assumptions

Given: A proper homologically simply connected complex domain $Ω ⊊ C$ and $z_{0} \in Ω$ .

[L1]

The extremal family $F (Ω, z_{0})$ is nonempty (A proper homologically simply connected plane domain has a bounded univalent competitor).

[L2]

Every $f \in F (Ω, z_{0})$ satisfies $f (z_{0}) = 0$ and $f^{'} (z_{0}) > 0$ (The extremal family of disc-valued univalent maps fixing a basepoint).

[L3]

Cauchy estimates bound derivatives from a modulus bound on a larger concentric circle (Cauchy estimates on a smaller concentric disc).

Proof

technique · direct

1.1

Fact [L1] gives at least one map in $F (Ω, z_{0})$ , so the derivative set $E$ is nonempty. Fact [L2] makes every element of $E$ strictly positive.

L1L2given

1.2

Choose $ρ > 0$ with $\overline{D (z_{0}, ρ)} \subset Ω$ . If $f \in F (Ω, z_{0})$ , then $∣ f ∣ \leq 1$ on $D (z_{0}, ρ)$ because $f (Ω) \subset D$ , so [L3] gives $∣ f^{'} (z_{0}) ∣ \leq 1 / ρ$ . Since [L2] makes $f^{'} (z_{0})$ positive real, this is the same as $f^{'} (z_{0}) \leq 1 / ρ$ .

L2L3givenchoose

2.1

Therefore $E \subset (0, 1 / ρ]$ , so $E$ has a finite supremum.

step 1.1step 1.2∎

Depends on

Used by

A maximizing sequence has a locally uniform limit with extremal derivative Lemma

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

Matthias Weber, Complex Analysis, §5.2 (standard reference, not scraped)
Walter Rudin, Real and Complex Analysis, Theorem 14.9 (standard reference, not scraped)