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

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A nonconstant locally uniform limit of univalent functions is univalent

Statement

Let $Ω$ be a complex domain, let $f_{n} : Ω \to C$ be univalent for every $n$ , and suppose $f_{n} \to f$ locally uniformly on $Ω$ . If $f$ is nonconstant, then $f$ is univalent.

Facts & Assumptions

Given: A complex domain $Ω$ , univalent maps $f_{n} : Ω \to C$ , and locally uniform convergence $f_{n} \to f$ to a nonconstant holomorphic limit.

[L1]

A univalent map is injective (Univalent holomorphic functions).

[L2]

A locally uniform limit of nowhere-zero holomorphic functions is either identically zero or nowhere zero (Hurwitz's zero-free limit theorem).

Proof

technique · direct

1.1

Fix $a \in Ω$ . For each $n$ , define $h_{n} (z) : = \frac{f_{n} (z) - f_{n} (a)}{z - a} (z \in Ω ∖ {a}) .$ Since each $f_{n}$ is injective by [L1], the function $h_{n}$ has no zeros on $Ω ∖ {a}$ . The removable singularity at $a$ is filled by $h_{n} (a) : = f_{n}^{'} (a)$ , so each $h_{n}$ is holomorphic and nowhere zero on $Ω$ .

L1givenalgebra

2.1

The functions $h_{n}$ converge locally uniformly to $h (z) : = {\begin{cases} \frac{f (z) - f (a)}{z - a}, & z \neq a, \\ f^{'} (a), & z = a, \end{cases}$ because $f_{n} \to f$ locally uniformly and derivatives converge locally uniformly as well. Fact [L2] therefore makes $h$ either identically zero or nowhere zero.

L2step 1.1algebra

3.1

Since $f$ is nonconstant, the function $h$ is not identically zero. Hence step 2.1 makes $h$ nowhere zero. If $f (z) = f (a)$ , then $h (z) = 0$ unless $z = a$ , so necessarily $z = a$ . As $a$ was arbitrary, $f$ is injective and therefore univalent by [L1].

L1step 2.1discharge-construct∎

Depends on

Used by

The extremal limit is univalent Lemma

Dependency tree · two levels

4 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, Theorem 5.2.6 (standard reference, not scraped)
Walter Rudin, Real and Complex Analysis, Theorem 14.9 (standard reference, not scraped)