Preimages of One-Point Sets of Bloch and Normal Functions

Abstract

We call a sequence {zn} in the open unit disk Δalmost thin if \({\mathop {\lim \sup }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 1,\) where Π(zn) is the product of the pseudo-hyperbolic distances between zn and the other elements in the sequence. We call the sequence thick if \({\mathop {\lim }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 0.\) In this paper, we use elementary complex variables methods to prove a sharper form of a characterization of the symbols of the isometric composition operators on the Bloch space proved by Martin and Vukotic in terms of the preimages of one-point sets. Specifically, we show that a bounded analytic function φ from Δ into itself has Bloch semi-norm equal to one (which is the largest possible) if and only if it is a conformal automorphism of Δ or if the preimage of each point of Δ contains a sequence along which the hyperbolic derivative of φ approaches one (condition (M’)). In particular, the preimage of each point of Δ is an almostthin sequence. Using similar techniques we also show that a bounded analytic function f is in the little Bloch space if and only if for each \(a\in \mathbb{C}\) such that \(f^{-1}(a)=\{z_{n}:n\in \mathbb{N}\}\) is infinite $$ {\mathop {\lim }\limits_{n \to \infty } }\left(1 - |z_{n} |^{2} \right)|f'{\left( {z_{n} } \right)}| = 0. $$ Consequently, an infinite Blaschke product whose left composition by a conformal automorphism of Δ is always a Blaschke product belongs to the little Bloch space if and only if the preimage of every point in Δ is a thick sequence. We give an example to show that this characterization does not extend to unbounded Bloch functions. We also prove an analogous characterization of the little normal functions. Finally, we give a sufficient condition on a holomorphic self-map of the polydisk Δn to be the symbol of an isometric composition operator on the Bloch space of Δn and discuss a higher-dimensional analogue of condition (M’).