The function f is called a Bloch function if it is analytic in the unit disk D and satisfies one of the following equivalent [8] conditions: 

(i)

(1 - |z|2)|f′(z)| is bounded in D;




(ii)

the Riemann image surface over ℂ contains no arbitrarily large schlicht disks;




(iii)

f = c log g′ where g is univalent in D and c is a suitable constant.

On the radial limits of functions with Hadamard gaps.

On the boundary behavior of Bloch functions

Sets with zdhd and zhd are defined. Finite unions of I 0 sets have zdhd. A “Hadamard gap” theorem holds for sets with zhd.

Sets of Zero Discrete Harmonic Density

In this section we recall three estimates due to J. Bourgain, Ch. Pommerenke and A. Beurling respectively. In section 2 the construction of stopping time Lipschitz domains is based on Pommerenke’s inequality. In section 3 the selection of good directions e is based on the result of J. Bourgain and estimates for harmonic measure due to A. Beurling. We first discuss Bourgain’s inequality from [B]. For e ∈ T we let Γα be the collection of curves γ which admit the following parametrization. For 0 < r < 1, γ(r) = ree where |θ(r)| < C(1− r) and |θ′(r)| < C. We fix a non-negative, harmonic function h in D and we let K be an interval in T. Then the following result was proven in [B].

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1997/0004/0003/MRL-1997-0004-0003-a008.pdf

Radial variation of Bloch functions

Sidon sets defined. Basic properties and characterizations. Examples. Quasi-independent and Rider sets. Λ(p) sets. Decompositions of Sidon sets.

Sidon Sets: Introduction and Decomposition Properties

Criteria are given for the union of I 0(U)sets to be I 0(U). Every I 0 set is a finite union of I 0(U)sets with bounded length. Criteria for an I 0 set to be a finite union of RI 0 (or FZI 0) sets are also given.

Unions and Decompositions of I0(U) Sets

https://projecteuclid.org/download/pdf_1/euclid.mmj/1029003128

On the radial limits of functions with Hadamard gaps.

Citations

On the boundary behavior of Bloch functions

Sets of Zero Discrete Harmonic Density

Radial variation of Bloch functions

Sidon Sets: Introduction and Decomposition Properties

Unions and Decompositions of I0(U) Sets

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