TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Bloch's theorem

Statement

If $f$ is holomorphic on $D$ and $∣ f^{'} (0) ∣ = 1$ , then

$β (f) \geq \frac{1}{48} .$

In particular, $B > 0$ .

Facts & Assumptions

Given: A holomorphic map $f : D \to C$ with $∣ f^{'} (0) ∣ = 1$ .

[L1]

The maximizing-point rescaling lemma produces a normalized map $g$ on $D$ with $∣ g^{'} ∣ \leq 2$ and $r ∣ f^{'} (z_{0}) ∣ \geq 1 / 4$ (Maximizing-point rescaling produces a normalized map with uniformly bounded derivative).

[L2]

Such a normalized map is univalent on $D (0, 1 / 6)$ and covers $D (0, 1 / 12)$ there (Controlled derivative oscillation forces injectivity on a fixed subdisc).

Proof

technique · direct

1.1L1givenchoose

Apply [L1] with $R = 1 / 2$ to obtain $z_{0} \in D (0, 1 / 2)$ , a radius $r > 0$ , and a normalized rescaling $g (w) = (f (z_{0} + r w) - f (z_{0})) / (r f^{'} (z_{0}))$ with $∣ g^{'} (w) ∣ \leq 2$ on $D$ and $r ∣ f^{'} (z_{0}) ∣ \geq 1 / 4$ .

2.1L2step 1.1constructalgebra

By [L2], the restriction of $g$ to $D (0, 1 / 6)$ is univalent and its image contains $D (0, 1 / 12)$ . Take the inverse image of that round disc under this univalent restriction and then undo the affine source and target normalizations. This gives a subdomain on which $f$ maps univalently onto $D ⁣ (f (z_{0}), r ∣ f^{'} (z_{0}) ∣ / 12)$ . Since step 1.1 gives $r ∣ f^{'} (z_{0}) ∣ \geq 1 / 4$ , this radius is at least $1 / 48$ .

3.1step 2.1∎

Thus $β (f) \geq 1 / 48$ . Taking the infimum over all normalized $f$ gives $B \geq 1 / 48 > 0$ .

Depends on

Used by

The elementary Bloch proof on this page yields the explicit lower bound 1/48 Example
Landau's theorem Theorem
Schottky's theorem Theorem

Dependency tree · two levels

7 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 7.4.2 (standard reference, not scraped)