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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

Not every compact set is conformally removable Counterexample
Analytic Hardy spaces on the unit disc Definition
Blaschke factors and Blaschke products Definition
Inner, singular inner and outer functions Definition
Quasisymmetric homeomorphisms of the line and circle Definition
The extremal family of disc-valued univalent maps fixing a basepoint Definition
The modular group and its action on the upper half-plane Definition
The modular lambda function Definition
The Poincare metric and distance on the unit disc Definition
The welding homeomorphism of a Jordan curve Definition
A genus-two compact surface gives a cocompact Fuchsian group Example
An irregular puncture does not force the Green kernel to vanish Example
Finite Blaschke products Example
Green kernel of the disc at a nonzero pole Example
Harmonic measure of an arc of the unit circle Example
Hyperbolic distances and geodesics in disc and half-plane Example
Inner-outer factorization of a rational function with one interior zero Example
Power maps, endpoint distortion, and a non-Möbius quasisymmetric circle map Example
The identity welding of the round circle Example
The Möbius ambiguity in conformal welding Example
The upper half-plane Bergman kernel by biholomorphic transport Example
The upper half-plane Poisson boundary density Example
Poisson-Jensen inequality for Hardy functions Lemma
Riemann maps of Jordan domains extend to homeomorphisms of the closures Lemma
Round circles and straight lines are conformally removable Lemma
The Ahlfors-Beurling extension formula for quasisymmetric maps of the line Lemma
The Lipschitz formula for the reciprocal-power sums Lemma
The transformation law of the weight-two Eisenstein series E₂ Lemma
Blaschke factors are automorphisms of the disc Theorem
Bounded turning, quasiconformal images of the circle, and quasiconformal reflections Theorem
Every quasisymmetric circle homeomorphism is a conformal welding Theorem
Green kernel of a simply connected plane domain from a Riemann map Theorem
Jacobi theta triple product and nonvanishing of the theta constant Theorem
Schwarz lemma with the equality cases Theorem
The Beurling–Ahlfors extension theorem for circles and lines Theorem
The q-expansion principle at the cusp Theorem

Dependency tree · two levels

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Sources

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