DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-29

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The unit disc, the upper half-plane, and Blaschke factors

Definition

Fix the unit disc and the upper half-plane

$D : = {z \in C : ∣ z ∣ < 1}, H : = {z \in C : Im z > 0},$

using modulus and imaginary part of Real and imaginary parts, complex conjugation, and modulus.

For $a \in D$ define the Blaschke factor

$φ_{a} (z) : = \frac{a - z}{1 - \overline{a} z} .$

For every $z$ with $∣ z ∣ \leq 1$ , one has $∣ \overline{a} z ∣ \leq ∣ a ∣ < 1$ , so the denominator never vanishes on the closed unit disc. Thus $φ_{a}$ is holomorphic on an open disc containing the closed unit disc and in particular on $D$ . Direct substitution records

$φ_{a} (0) = a, φ_{a} (a) = 0.$

Depends on

Real and imaginary parts, complex conjugation, and modulus

Used by

The Poincare metric and distance on the unit disc Definition
Blaschke factors are automorphisms of the disc Theorem
Schwarz lemma with the equality cases Theorem

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

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §2 (standard reference, not scraped)