Remarks on one-component inner functions

Abstract

A one-component inner function Θ is an inner function whose level set Ω Θ (ε) = {z ∈ D : |Θ(z)| < ε} is connected for some ε ∈ (0, 1). We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for 0 < p < ∞, the derivative of a one-component inner function Θ is a member of the Hardy space H p if and only if Θ ′′ belongs to the Bergman space A p p−1 , or equivalently Θ ′ ∈ A 2p p−1. 1. Examples of one-component inner functions Let D be the open unit disc of the complex plane C. A bounded and analytic function in D is an inner function if it has unimodular radial limits almost everywhere on the boundary T of D. In this note, we study so-called one-component inner functions [14], which are inner functions Θ whose level set Ω Θ (ε) = {z ∈ D : |Θ(z)| < ε} is connected for some ε ∈ (0, 1). In particular, Blaschke products in this class are of interest. For a given sequence {z n } ⊂ D \ {0} satisfying n (1 − |z n |) < ∞, the Blaschke product with zeros {z n } is defined by B(z) = n |z n | z n z n − z 1 − z n z , z ∈ D. Here each zero z n is repeated according to its multiplicity. In addition, we assume that {z n } is ordered by non-decreasing moduli. Recently several authors have studied one-component inner functions in the context of model spaces and operator theory; see for instance [6, 8, 9, 10]. In addition, Aleksandrov's paper [5], which contains several characterizations for one-component inner functions, is worth mentioning. These references do not offer any concrete examples of infinite one-component Blaschke products; even though, reference [5] offers tools for this purpose. In recent paper [13] by Cima and Mortini, one can find some examples. However, all one-component Blaschke products constructed in [13] have some heavy restrictions. Roughly speaking, zeros of all of them are at least uniformly separated. Recall that {z n } ⊂ D is called uniformly separated if inf n∈N k =n z k − z n 1 − z k z n > 0.

A characterization of one‐component inner functions

“…As a byproduct he proved also a strong form of the Schwarz–Pick lemma for inner functions in . Using Aleksandrov's descriptions, Cima, Mortini and the second author constructed some concrete examples of one‐component inner functions [10, 11, 23]. In particular singular inner functions associated to a finite sum of weighted Dirac masses are one‐component and thin Blaschke products are not in .…”

“…For and , the Bergman space consists of those analytic functions in such that where is the Lebesgue area measure on . Note that [23, Theorem 10] implies the inclusion Hence using Corollary 5 one can construct one‐component inner functions whose derivative does not belong to certain Bergman spaces.…”

A Characterization of One-component Inner Functions

“…As a byproduct he proved also a strong form of the Schwarz-Pick lemma for inner functions in I c . Using Aleksandrov's descriptions, J. Cima, R. Mortini and the second author constructed some concrete examples of one-component inner functions [10,11,23]. In particular singular inner functions associated to a finite sum of weighted Dirac masses are one-component and thin Blaschke products are not in I c .…”