Open Access

One-component inner functions

Complex Analysis and its Synergies20173:2

https://doi.org/10.1186/s40627-016-0008-8

Received: 7 September 2016

Accepted: 16 December 2016

Published: 25 January 2017

Abstract

We explicitly unveil several classes of inner functions u in \(H^\infty \) with the property that there is \(\eta \in ]0,1[\) such that the level set \(\Omega _u(\eta ):=\{z\in {\mathbb D}: |u(z)|<\eta \}\) is connected. These so-called one-component inner functions play an important role in operator theory.

Keywords

Inner functionsInterpolating Blaschke productsConnected componentsLevel sets

Mathematics Subject Classification

Primary 30J10Secondary 30J05

1 Background

Definition 1

An inner function u in \(H^\infty \) is said to be a one-component inner function if there is \(\eta \in ]0,1[\) such that the level set (also called sublevel set or filled level set) \(\Omega _u(\eta ):=\{z\in {\mathbb D}: |u(z)|<\eta \}\) is connected.

One-component inner functions, the collection of which we denote by \(\mathfrak I_c,\) were first studied by Cohn [10] in connection with embedding theorems and Carleson measures. It was shown that [for instance, [10], p. 355] arclength on \(\{z\in {\mathbb D}: |u(z)|=\varepsilon \}\) is such a measure whenever
$$\begin{aligned} \Omega _u(\eta )=\{z\in {\mathbb D}: |u(z)|<\eta \} \end{aligned}$$
is connected and \(\eta<\varepsilon <1\).
A thorough study of the class \(\mathfrak I_c\) was given by Aleksandrov [1] who showed the interesting result that \(u\in \mathfrak I_c\) if and only if there is a constant \(C=C(u)\) such that for all \(a\in {\mathbb D}\)
$$\begin{aligned} \sup _{z\in {\mathbb D}}\left| \frac{1-\overline{u(a)} u(z)}{1-\overline{a} z} \right| \le C \frac{1-|u(a)|^2}{1-|a|^2}. \end{aligned}$$
Many operator-theoretic applications are given in [13, 7]. In our paper here, we are interested in explicit examples, which are somewhat lacking in the literature. For example, if S is the atomic inner function, which is given by
$$\begin{aligned} S(z)=\exp \left( - \frac{1+z}{1-z}\right) , \end{aligned}$$
then all level sets \(\Omega _S(\eta )\), \(0<\eta <1\) are connected, because these sets coincide with the disks
$$\begin{aligned} D_\eta :=\left\{ z\in {\mathbb D}:\left| z- \frac{L}{L+1}\right| <\frac{1}{L+1}\right\} ,\quad L:= \log \frac{1}{\eta }, \end{aligned}$$
(1)
which are tangential to the unit circle at \(p=1\).

The scheme of our note here is as follows: in Sect. 2, we prove a general result on level sets which will be the key for our approach to the problem of unveiling classes of one-component inner functions. Then in Sect. 3, we first present several examples with elementary geometric/function theoretic methods and then we use Aleksandrov’s criterion to achieve this goal. For instance, we prove that \(BS, B\circ S\), and \(S\circ B\) are in \(\mathfrak I_c\) whenever B is a finite Blaschke product. Considered are also interpolating Blaschke products. It will further be shown that, under the supremum norm, \(\mathfrak I_c\) is an open subset of the set of all inner functions and multiplicatively closed. In the final section, we give counterexamples.

2 Level sets

We first begin with a topological property of the class of general level sets. Although statement (1) is “well known” (the earliest appearance seems to be in [26, Theorem VIII,  31]), we could nowhere locate a proof. The argument that the result is a simple and direct consequence of the maximum principle is, in our viewpoint, not tenable.

Lemma 2

Given a non-constant inner function u in \(H^\infty \) and \(\eta \in \;]0,1[,\) let \(\Omega :=\Omega _u(\eta )=\{z\in {\mathbb D}: |u(z)|<\eta \}\) be a level set. Suppose that \(\Omega _0\) is a component (=maximal connected subset) of \(\Omega. \) Then
  1. (1)

    \(\Omega _0\) is a simply connected domain; that is, \({ \mathbb C}\setminus \Omega _0\) has no bounded components. 1

     
  2. (2)

    \(\it{\inf} _{\Omega _0} |u|=0\).

     

Proof

We show that item (1) holds for every holomorphic function f in \({\mathbb D}\); that is, if \(\Omega _0\) is a component of the level set \(\Omega _f(\eta )\), \(\eta >0\), then it is a simply connected domain.2 Note that each component \(\Omega _0\) of the open set \(\Omega _f(\eta )\) is an open subset of \({\mathbb D}\). We may assume that \(\eta \) is chosen so that \(\{z\in {\mathbb D}: |f(z)|=\eta \}\not =\emptyset \).

Suppose that, on the contrary, D is a bounded component of \({ \mathbb C}\setminus \Omega _0\). Note that D is closed in \({ \mathbb C}\). Then, necessarily, D is contained in \({\mathbb D}\), because the unique unbounded complementary component of \(\Omega _0\) contains \(\{z\in { \mathbb C}: |z|\ge 1\}\). Hence, D is a compact subset of \({\mathbb D}\). Let \(G:=\Omega _0^*\) be the simply connected hull of \(\Omega _0:\) the union of \(\Omega _0\) with all bounded complementary components of \(\Omega _0\). Note that G is open because it coincides with the complement of the unique unbounded complementary component of \(\Omega _0\). Then, by definition of the simply connected hull, \(D\subseteq G\). Now if H is any bounded complementary component of \(\Omega _0\) then (as it was the case for D), H is a compact subset of \({\mathbb D}\) and so \(\partial H\subseteq {\mathbb D}\). Moreover,
$$\begin{aligned} \partial H\subseteq \partial \Omega _0. \end{aligned}$$
(2)
In fact, given \(z_0\in \partial H\), let U be a disk centered at \(z_0\). Then \(U\cap \Omega _0\not =\emptyset \), since otherwise \(U\cup H\) would be a connected set strictly bigger than H and contained in the complement of \(\Omega _0:\) a contradiction to the maximality of H. Since \(z_0\in \partial H\subseteq H\subseteq { \mathbb C}\setminus \Omega _0,\) we conclude that \(z_0\in \partial \Omega _0.\)
Now \(\partial H\subseteq \partial \Omega _0\) and \(\Omega _0\subseteq \Omega _f(\eta )\) imply that \(|f|\le \eta \) on \(\partial H\), and so, by the maximum principle, \(|f|\le \eta \) on H. Consequently, \(|f|\le \eta \) on G. By the local maximum principle, \(|f|<\eta \) on G. Since \(\partial D\subseteq D\subseteq G\),
$$\begin{aligned} |f|<\eta \,\text{ on }\, \partial D. \end{aligned}$$
(3)
On the other hand,
$$\begin{aligned} \partial D \mathop \subseteq \limits _{}^{(2)} \partial \Omega _0\cap {\mathbb D}\subseteq \{z\in {\mathbb D}: |f(z)|=\eta \}. \end{aligned}$$
(4)
Note that the second inclusion follows from the fact that if \(|f(z_0)|<\eta \) for \(z_0\in \partial \Omega _0\cap {\mathbb D}\), then \(\Omega _0\) would no longer be a maximal connected subset of \(\Omega _f(\eta )\). Hence, \(|f|=\eta \) on \(\partial D\). This is a contradiction to (3). Thus, \(\Omega _0\) is a simply connected domain.
(2) If \(\overline{\Omega }_0\subseteq {\mathbb D}\), then, due to \(\partial \Omega _0\subseteq \{z\in {\mathbb D}: |u(z)|=\eta \}\), we obtain from the minimum principle that u must have a zero in \(\Omega _0.\) Now let \(E:=\overline{\Omega }_0\cap \partial {\mathbb D}\not =\emptyset \). In view of achieving a contradiction, suppose that u is bounded away from zero in \(\Omega _0\). Then 1 / |u| is subharmonic and bounded in \(\Omega _0\) and
$$\begin{aligned} \limsup _{\begin{array}{c} \xi \rightarrow x\\ x\in \partial \Omega _0\setminus E \end{array}}|u(\xi )|^{-1}=\eta ^{-1}. \end{aligned}$$
Since u is an inner function, E has linear measure zero (by [5, Theorem 4.2]). The maximum principle for subharmonic functions with a few exceptional points (here on the set E; see [6] or [12]) now implies that \(|u|^{-1}\le \eta ^{-1}\) on \(\Omega _0.\) But \(|u|<\eta \) on \(\Omega \) is a contradiction. We conclude that \(\inf _{\Omega _0}|u|=0\). \(\square \)

Lemma 3

[10] Let u be an inner function. Then the connectedness of \(\Omega _u(\eta )\) implies the one of \(\Omega _u(\eta ')\) for every \(\eta '>\eta \).

Proof

Because \(\Omega _u(\eta )\) is connected and \(\Omega _u(\eta )\subseteq \Omega _u(\eta '),\) \(\Omega _u(\eta )\) is contained in a unique component \(U_1(\eta ')\) of \(\Omega _u(\eta ').\) Suppose that \(U_0(\eta ')\) is a second component of \(\Omega _u(\eta ').\) Then \(|u|\ge \eta \) on \(U_0(\eta '),\) because \(U_0(\eta ')\) is disjoint with \(U_1(\eta ')\) and hence with \(\Omega _u(\eta ).\) By Lemma 2 though, \(\inf _{U_0(\eta ')} |u|=0;\) a contradiction. Thus \(\Omega _u(\eta ')\) is connected. \(\square \)

3 Explicit examples of one-component inner functions

Let
$$\begin{aligned} \rho (z,w)=\left| \frac{z-w}{1-\overline{z}w}\right| \end{aligned}$$
be the pseudohyperbolic distance of z to w in \({\mathbb D}\) and
$$\begin{aligned} D_\rho (z_0,r)=\{z\in {\mathbb D}: \rho (z,z_0)<r\} \end{aligned}$$
the associated disks, \(0<r<1\). Here is a first class of examples of functions in \(\mathfrak I_c.\) Although the next Proposition must be known (in view of Aleksandrov’s criterion [1]), see Theorem 15 below), we include a simple geometric proof for the reader’s convenience.

Proposition 4

Let B be a finite Blaschke product. Then \(B\in \mathfrak I_c.\)

Proof

Denote by \(z_1,\dots ,z_N\) the zeros of B, multiplicities included. Let \(\eta \in \;[0,1]\) be chosen so close to 1 that \(G:=\bigcup _{n=1}^N D_\rho (z_n,\eta )\) is connected (for example by choosing \(\eta \) so that \(z_j\in D_\rho (z_1,\eta )\) for all j). Now
$$\begin{aligned} G\subseteq \{z\in {\mathbb D}: |B(z)|<\eta \}=\Omega _B(\eta ), \end{aligned}$$
because \(z\in G\) implies that for some n,
$$\begin{aligned} |B(z)|=\rho (B(z), B(z_n))\le \rho (z,z_n)<\sigma . \end{aligned}$$
Since G is connected, there is a unique component \(\Omega _1\) of \(\Omega \) containing G. In particular, \(Z(B)\subseteq G\subseteq \Omega _1.\) If, in view of achieving a contradiction, we suppose that \(\Omega :=\Omega _B(\eta )\) is not connected, there is a component \(\Omega _0\) of \(\Omega \) which is disjoint with \(\Omega _1,\) and so with G. In particular,
$$\begin{aligned} \rho (z, Z(B))\ge \sigma \;\; \text {for every}\;\; z\in \Omega _0. \end{aligned}$$
(5)
Since \(\overline{\Omega }_0 \subseteq \overline{\Omega _B(\eta )}\subseteq {\mathbb D},\) and \(|B|=\eta \) on \(\partial \Omega _0,\) we deduce from the minimum principle that \(\Omega _0\) contains a zero of B; a contradiction. \(\square \)
We now generalize this result to a class of interpolating Blaschke products. Recall that a Blaschke product b with zero set/sequence \(\{z_n:n\in { \mathbb N}\}\) is said to be an interpolating Blaschke product if \(\delta (b):=\inf (1-|z_n|^2)|b'(z_n)|>0\). If b is an interpolating Blaschke product then, for small \(\varepsilon \), the pseudohyperbolic disks
$$\begin{aligned} D_\rho (z_n,r)=\{z\in {\mathbb D}: \rho (z,z_n)<\varepsilon \} \end{aligned}$$
are pairwise disjoint. Moreover, by Hoffman’s Lemma (see below and also [19]), for any \(\eta \in ]0,1[\), b is bounded away from zero on \(\{z\in {\mathbb D}: \rho (z,Z(b))\ge \eta \}\).

Theorem 5

(Hoffman’s Lemma) ([18, p. 86, 106] and [13, p. 404, 310]) Let \(\delta , \eta \) and \(\epsilon \) be real numbers, called Hoffman constants, satisfying \(0<~\delta <~1\), \( 0< \eta < (1-\sqrt{1-\delta ^2})/ \delta \) , (that is, \(0< \eta < \rho (\delta ,\eta )\) ) and
$$\begin{aligned} 0< \varepsilon < \eta \frac{\delta - \eta }{1- \delta \eta }. \end{aligned}$$
If B is any interpolating Blaschke product with zeros \(\{z_n:n\in { \mathbb N}\}\) such that
$$\begin{aligned} \delta (B) = \inf _{n \in {\mathbb {N}}} (1- | z_n |^2) | B'(z_n)| \ge \delta , \end{aligned}$$
then
  1. (1)

    The pseudohyperbolic disks \(D_\rho (a,\eta )\) for \(a\in Z(B)\) are pairwise disjoint.

     
  2. (2)
    The following inclusions hold:
    $$\begin{aligned} \{z \in {\mathbb D}: |B(z)|< \varepsilon \} \subseteq \{z \in {\mathbb D}: \rho (z, Z(B))< \eta \} \subseteq \{z \in {\mathbb D}: |B(z)| < \eta \}. \end{aligned}$$
     

A large class of interpolating Blaschke products which are one-component inner functions now is given in the following result:

Theorem 6

Let b be an interpolating Blaschke product with zero set \(\{z_n:n\in { \mathbb N}\}\). Suppose that for some \(\sigma \in \;]0,1[\) the set
$$\begin{aligned} G:=\bigcup _n D_\rho (z_n,\sigma ) \end{aligned}$$
is connected. Then b is a one-component inner function. This holds in particular, if \(\rho (z_n,z_{n+1})<\sigma <1\) for all n: for example if \(z_n =1-2^{-n}\).

Proof

As in the proof of Proposition 4
$$\begin{aligned} G\subseteq \{z\in {\mathbb D}: |b(z)|<\sigma \}=:\Omega . \end{aligned}$$
Since G is assumed to be connected, there is a unique component \(\Omega _1\) of \(\Omega \) containing G. In particular, \(Z(b)\subseteq G\subseteq \Omega _1\). Now, if we suppose that \(\Omega \) is not connected, then there is a component \(\Omega _0\) of \(\Omega \) which is disjoint with \(\Omega _1\), and so with G. In particular,
$$\begin{aligned} \rho (z, Z(b))\ge \sigma \;\; \text {for every}\;\; z\in \Omega _0. \end{aligned}$$
(6)
Let \(\delta :=\delta (b)\),
$$\begin{aligned} 0< \eta < \min \{(1-\sqrt{1-\delta ^2})/ \delta , \sigma \}, \end{aligned}$$
$$\begin{aligned} 0< \varepsilon < \eta \frac{\delta - \eta }{1- \delta \eta }. \end{aligned}$$
By Lemma 2, \(\inf _{\Omega _0}|b|=0\). Thus, there is \(z_0\in \Omega _0\) be so that \(|b(z_0)|<\varepsilon \). We deduce from Hoffman’s Lemma (Theorem 5) that \(\rho (z_0,Z(b))<\eta <\sigma \). This is a contradiction to (6). We conclude that \(\Omega \) must be connected. It is clear that the condition \(\rho (z_n,z_{n+1})<\sigma \) for every n implies that \(\bigcup _n D_\rho (z_n,\sigma )\) is connected. For the rest, just note that
$$\begin{aligned} \rho (1-2^{-n}, 1-2^{-n-1})=\frac{2^{-n}-2^{-n-1}}{2^{-n}+2^{-n-1}+2^{-n}2^{-n-1}}=\frac{1}{3 +2^{-n}}. \end{aligned}$$
\(\square \)

Corollary 7

Let B be a Blaschke product with increasing real zeros \(x_n\) satisfying
$$\begin{aligned} 0<\eta _1\le \rho (x_n, x_{n+1})\le \eta _2<1. \end{aligned}$$
Then \(b\in \mathfrak I_c\).

Proof

Just use Theorem 6 and the fact that by the Vinogradov–Hayman–Newman theorem, B is interpolating if and only if
$$\begin{aligned} \sup _n\frac{1-x_{n+1}}{1-x_n}\le s<1 \end{aligned}$$
or equivalently
$$\begin{aligned} \inf _n\rho (x_n,x_{n+1})\ge r>0. \end{aligned}$$
\(\square \)

Using a result of Kam-Fook Tse [25], telling us that a sequence \((z_n)\) of points contained in a Stolz angle (or cone) \(\{z\in {\mathbb D}: |1-z|< C (1-|z|)\}\) is interpolating if and only if it is separated (meaning that \(\inf _{n\not =m}\rho (z_n,z_m)>0\)), we obtain the following Corollary

Corollary 8

Let B be a Blaschke product the zeros \((z_n)\) of which are contained in a Stolz angle and are separated. Suppose that \(\rho (z_n,z_{n+1})\le \eta <1\). Then \(B\in \mathfrak I_c.\)

Similarly, using a result by M. Weiss [27, Theorem 6] and its refinement in [4, Theorem B], we also obtain the following assertion for sequences that may be tangential at 1 (see also Wortman [28]):

Corollary 9

Let B be a Blaschke product the zeros \(z_n=r_ne^{i\theta _n}\) of which satisfy:
$$\begin{aligned} r_n<r_{n+1}, ~ \theta _{n+1}< \theta _n, \end{aligned}$$
$$\begin{aligned} r_n\nearrow 1,~ \theta _n\searrow 0, \end{aligned}$$
$$\begin{aligned} 0<\eta _1\le \rho (z_n,z_{n+1})\le \eta _2<1. \end{aligned}$$
(7)
Then B is an interpolating Blaschke product contained in \(\mathfrak I_c\). This holds in particular if the zeros are located on a convex curve in \({\mathbb D}\) with endpoint 1 and satisfying (7).

Other classes of this type can be deduced from [14]. Here are two explicit examples of interpolating Blaschke products in \(\mathfrak I_c\) the zeros of which are given by iteration of the zero of a hyperbolic, respectively, parabolic automorphism of \({\mathbb D}\). These functions appear, for instance, in the context of isometries on the Hardy space \(H^p\) (see [8]).

Example 10

\(\bullet \)   Let \(\displaystyle \varphi (z)=\frac{z-1/2}{1- (1/2) z}\). Then \(\varphi \) is an hyperbolic automorphism with fixed points \(\pm 1\). If \(\varphi _j:=\underbrace{\varphi \circ \cdots \circ \varphi }_{j{\text{-}}\mathrm{times}}\), then \(\varphi _j\in \mathrm{Aut}({\mathbb D})\) and vanishes exactly at the point
$$\begin{aligned} x_j:=\frac{3^j-1}{3^j+1}= 1-\frac{2}{3^j+1}. \end{aligned}$$
This can readily be seen using that \(x_{j+1}=\varphi ^{-1}(x_j)\) and
$$\begin{aligned} \varphi _{j+1}(z) = (\varphi _ j\circ \varphi )(z)=\frac{z- \frac{\frac{1}{2}+x_j}{1+ \frac{1}{2}x_j}}{1-z \frac{\frac{1}{2}+x_j}{1+ \frac{1}{2}x_j}}. \end{aligned}$$
Since
$$\begin{aligned} \rho (x_j,x_{j+1})=\frac{3^{j+1}-3^j}{3^{j+1}+ 3^j}=\frac{1}{2}, \end{aligned}$$
we deduce from Corollary 7 that the Blaschke product
$$\begin{aligned} B(z):=\prod _{j=1}^\infty \frac{x_j-z}{1-x_j z} \end{aligned}$$
associated with these zeros is in \(\mathfrak I_c.\)
\(\bullet \) Let \(\sigma \in \mathrm{Aut}({\mathbb D})\) and \(\tau =\sigma \circ \varphi \circ \sigma ^{-1}.\) Then \(\tau \) is also a hyperbolic automorphism fixing the points \(\sigma (\pm 1)\), and where \(\xi :=\sigma (1)\) is the Denjoy–Wolff point with \(\tau '(\xi )<1\). The zeros of the n-th iterate \(\tau _n\) of \(\tau \) are given by
$$\begin{aligned} z_n=\tau _n^{-1}(0)=(\sigma \circ \varphi ^{-1}_n\circ \sigma ^{-1})(0). \end{aligned}$$
By the grand iteration theorem [23, p.78], since 1 is an attracting fixpoint with \((\varphi ^{-1})'(1)= 1/3<1\), the sequence \((\varphi ^{-1}_n(\sigma ^{-1}(0)))\) converges nontangentially to 1. Hence, the points \(z_n\) are located in a cone with cusp at \(\xi \). Moreover, if \(n>k\) and \(a=\sigma ^{-1}(0)\),
$$\begin{aligned} \rho (z_n,z_k)= \, &{} \rho \big ( (\varphi ^{-1}_n\circ \sigma ^{-1})(0), (\varphi ^{-1}_k\circ \sigma ^{-1})(0)\big )\\= \, & {} \rho \big (\varphi ^{-1}_{n-k}(a), a\big ) \end{aligned}$$
Thus, \(\rho (z_n,z_{n+1})=\rho (\varphi (a), a)\) for all n and \(\inf _{n\not =k}\rho (z_n,z_k)>0\). Now \((z_n)\) is a Blaschke sequence3 ([23, Ex. 6, p. 85]); in fact, use d’Alembert’s quotient criterion and observe that by the Denjoy–Wolff theorem,
$$\begin{aligned} \frac{1-|z_{n+1}|}{1-|z_n|}= \frac{1-|\tau ^{-1}(z_n)|}{1-|z_n|}\rightarrow (\tau ^{-1})'(\xi )<1. \end{aligned}$$
Hence, by Corollary 8, \((z_n)\) is an interpolating sequence (see also [11, p.80]) and the associated Blaschke product \(b=\prod _{n=1}^\infty e^{i\theta _n} \tau _n\) belongs to \(\mathfrak I_c\) (here \(\theta _n\) is chosen so that the nth Blaschke factor is positive at the origin).
\(\bullet \)   Let \(\displaystyle \psi (z)=i\; \frac{z- \frac{1+i}{2}}{1- \frac{1-i}{2}\; z}\). Then \(\psi \) is a parabolic automorphism with attracting fixed point 1. The automorphism \(\psi \) is conjugated to the translation \(w\mapsto w+2\) on the upper half-plane (see Fig. 1) via the map \(M(z)= i (1+z)/(1-z)\) and \(\psi _n=M^{-1}\circ T_n\circ M\). The zeros of the n-th iterate \(\psi _n\) of \(\psi \) are given by
$$\begin{aligned} z_n=\frac{n}{n-i}; \end{aligned}$$
just use that \(z_n=(M^{-1}\circ T_n^{-1}\circ M)(0)\). These zeros satisfy \(\left| z_n-\frac{1}{2}\right| =\frac{1}{2}\) and of course also the Blaschke condition \(\sum _{n=1}^\infty 1-|z_n|^2<\infty \). Moreover,
$$\begin{aligned} \rho (z_n,z_{n+1})=\frac{1}{\sqrt{2}}. \end{aligned}$$
Thus, by, Corollary 9, the Blaschke product associated with these zeros is interpolating and belongs to \(\mathfrak I_c\).
Fig. 1

The parabolic automorphism

Proposition 11

Let B be a finite Blaschke product or an interpolating Blaschke product with real zeros clustering at \(p=1\). Then \(f:=BS\in \mathfrak I_c\).

Proof

  1. (i)

    Let B be a finite Blaschke product. Chose \(\eta \in \;[0,1]\) so close to 1 that the disk \(D_\eta \) in (1), which coincides with the level set \(\Omega _S(\eta )\), contains all zeros of B. Note that \(D_\eta =\Omega _S(\eta )\subseteq \Omega _f(\eta )\). Now \(\Omega _f(\eta )\) must be connected, since otherwise there would be a component \(\Omega _0\) of \(\Omega _f(\eta )\) disjoint from the component \(\Omega _1\) containing \(D_\eta \). But f is bounded away from zero outside \(D_\eta; \) hence, \(f=BS\) is bounded away from zero on \(\Omega _0.\) This is a contradiction to Lemma 2 (2).

     
  2. (ii)

    If B is an interpolating Blaschke product with zeros \((z_n)\), then, by Hoffman’s Lemma (Theorem 5), B is bounded away from zero outside \(R:=\bigcup D_\rho (z_n,\varepsilon )\) for every \(\varepsilon \in \;[0,1].\) Now, if the zeros of B are real, and bigger than \(-\sigma \) for some \(\sigma \in [0,1],\) this set R is contained in a cone with cusp at 1 and aperture-angle strictly less than \(\pi \) (for instance, [21]). Hence, R is contained in \(D_\eta \) for all \(\eta \) close to 1. Thus, as above, we can deduce that \(\Omega _{BS}(\eta )\) is connected.

     
\(\square \)

The previous result shows, in particular, that certain non one-component inner functions (for example a thin Blaschke product with positive zeros, see Corollary 21), can be multiplied by a one-component inner function into \(\mathfrak I_c\). In particular, \(uv\in \mathfrak I_c\) does not imply that u and v belong to \(\mathfrak I_c\). The reciprocal, though, is true: that is, \(\mathfrak I_c\) itself is stable under multiplication, as we proceed to show below.

Proposition 12

Let u, v be two inner functions in  \(\mathfrak I_c\). Then \(uv\in \mathfrak I_c.\)

Proof

Let \(\Omega _u(\eta )\) and \(\Omega _v(\eta ')\) be two connected level sets. Due to monotonicity (Lemma 3), and the fact that \(\bigcup _{\lambda \in [\lambda _0,1[} \Omega _f (\lambda )={\mathbb D}\), we may assume that \(\sigma \) satisfies
$$\begin{aligned} \max\, \{\eta ,\eta '\}\le \sigma <1 \end{aligned}$$
and is so close to 1 that \(0\in \Omega _u(\sigma )\cap \Omega _v(\sigma )\not =\emptyset \). Hence, \(U:=\Omega _u(\sigma )\cup \Omega _v(\sigma )\) is connected. Now
$$\begin{aligned} \Omega _u(\sigma )\cup \Omega _v(\sigma )\subseteq \Omega _{uv}(\sigma ). \end{aligned}$$
If we suppose that \(\Omega _{uv}(\sigma )\) is not connected, then there is a component \(\Omega _0\) of \(\Omega _{uv}(\sigma )\) which is disjoint from U. In particular, u and v are bounded away from zero on \(\Omega _0\). This contradicts Lemma 2 (2). Hence, \(\Omega _{uv}(\sigma )\) is connected and so \(uv\in \mathfrak I_c.\) \(\square \)

Theorem 13

The set of one-component inner functions is open inside the set of all inner functions (with respect to the uniform norm topoplogy).

Proof

Let \(u\in \mathfrak I_c\). Then, by Lemma 3, \(\Omega _u(\eta )\) is connected for all \(\eta \in [\eta _0,1[\). Choose \(0<\varepsilon <\min \{\eta , 1-\eta \}\) and let \(\Theta \) be an inner function with \(||u-\Theta ||<\varepsilon \). We claim that \(\Theta \in \mathfrak I_c,\) too. Toward this end, we note that
$$\begin{aligned} \Omega _{\Theta }(\eta -\varepsilon )\subseteq \Omega _u(\eta )\subseteq \Omega _\Theta (\eta +\varepsilon ), \end{aligned}$$
where \(0<\eta -\varepsilon<\eta +\varepsilon <1\). As usual, if we suppose that \(\Omega _{\Theta }(\eta +\varepsilon )\) is not connected, then there is a component \(\Omega _0\) of \(\Omega _{\Theta }(\eta +\varepsilon )\) which is disjoint from the connected set \(\Omega _u(\eta ),\) and hence disjoint with \(\Omega _{\Theta }(\eta -\varepsilon ).\) In other words, \(|\Theta |\ge \eta -\varepsilon >0\) on \(\Omega _0.\) This contradicts Lemma 2 (2). H \(\Omega _\Theta (\eta +\varepsilon )\) is connected and so \(\Theta \in \mathfrak I_c\). \(\square \)

Next we look at right-compositions of S with finite Blaschke products. We first deal with the case where \(B(z)=z^2\).

Example 14

The function \(S(z^2)\) is a one-component inner function.

Proof

Let \(\Omega _S(\eta )\) be the \(\eta \)-level set of S. Then, as we have already seen, this is a disk tangent to the unit circle at the point 1 (Fig. 2). We may choose \(0<\eta <1\) so close to 1 that 0 belongs to \(\Omega _S(\eta )\). Let \(U=\Omega _S(\eta )\setminus ]-\infty , 0].\) Then U is a simply connected slitted disk on which exists a holomorphic square root q of z. The image of U under q is a simply connected domain V in the semi-disk \(\{z: |z|<1, \mathrm{Re}\; z>0\}.\) Let \( V^*\) be its reflection along the imaginary axis. Then \(E:=\overline{V^*\cup V }\) is mapped by \(z^2\) onto the closed disk \(\overline{\Omega _S(\eta )}\). Then \(E{\setminus}\partial E\) coincides with \(\Omega _{S(z^2)}(\eta ).\) \(\square \)

Fig. 2

The level sets of \(S(z^2)\)

Using Aleksandrov’s criterion (see below), we can extend this by replacing \(z^2\) with any finite Blaschke product. Recall that the spectrum \(\rho (\Theta )\) of an inner function \(\Theta \) is the set of all boundary points \(\zeta \) for which \(\Theta \) does not admit a holomorphic extension—or equivalently, for which \(Cl(\Theta ,\zeta )=\overline{{\mathbb D}}\), where
$$\begin{aligned} Cl(\Theta ,\zeta )=\{w\in { \mathbb C}: \exists (z_n)\in {\mathbb D}^{ \mathbb N},~ \lim z_n=\zeta \, \text{ and }\, \lim \Theta (z_n)=w\} \end{aligned}$$
is the cluster set of \(\Theta \) at \(\zeta \) (see [13, p. 80]).

Theorem 15

(Aleksandrov) [1, Theorem 11 and Remark 2, p. 2915] Let \(\Theta \) be an inner function. The following assertions are equivalent:
  1. (1)

    \(\Theta \in \mathfrak I_c.\)

     
  2. (2)
    There is a constant \(C>0\) such that for every \(\zeta \in \mathbb T\setminus \rho (\Theta )\) we have
    $$\begin{aligned} i)~~~ |\Theta '' (\zeta )|\le C\; |\Theta ' (\zeta )|^2, \end{aligned}$$
    and
    $$\begin{aligned} ii)~~~ \liminf _{r\rightarrow 1} |\Theta (r\zeta )|<1 \, \text{ for } \text{ all }\, \zeta \in \rho (\Theta ). \end{aligned}$$
     

Note that, due to the above theorem, \(\Theta \in \mathfrak I_c\) necessarily implies that \(\rho (\Theta )\) has measure zero.

Proposition 16

Let B be a finite Blaschke product. Then \(S\circ B\in \mathfrak I_c.\)

Proof

Let us note first that \(\rho (S\circ B)=B^{-1}(\{1\})\). Since the derivative of B on the boundary never vanishes (due to
$$\begin{aligned} z\frac{B'(z)}{B(z)}=\sum _{n=1}^N \frac{1-|a_n|^2}{|a_n-z|^2}, ~~|z|=1, B(a_n)=0,) \end{aligned}$$
(8)
B is schlicht in a neighborhood of 1. The angle conservation law now implies that for \(\zeta \in B^{-1}(1)\) the curve \(r\mapsto B(r\zeta )\) stays in a Stolz angle at 1 in the image space of B. Hence \(\liminf _{r\rightarrow 1} S(B(r \zeta ))=0\) for \(\zeta \in \rho (S\circ B)\). Now let us calculate the following derivatives:
$$\begin{aligned} S'(z)=-S(z) \frac{2}{(1-z)^2}, \end{aligned}$$
$$\begin{aligned} S''(z)= S(z)\left[ \frac{4}{(1-z)^4}- \frac{4}{(1-z)^3}\right] , \end{aligned}$$
$$\begin{aligned} (S\circ B)'= (S'\circ B) B' \end{aligned}$$
$$\begin{aligned} (S\circ B)'' =(S''\circ B) B'^2 + (S'\circ B) B'' \end{aligned}$$
$$\begin{aligned} A:=\frac{(S\circ B)''}{[(S\circ B)']^2} &= {} \frac{S''\circ B}{(S'\circ B)^2}+ \frac{(S'\circ B)}{(S'\circ B)^2}\frac{B''}{B'^2}\\\nonumber &= {} \frac{S''\circ B}{(S'\circ B)^2}+ \frac{1}{S'\circ B}\frac{B''}{B'^2}. \end{aligned}$$
(9)
Hence, for \(\zeta \in \mathbb T\setminus \rho (S\circ B)\), \(|B(\zeta )|=1\) , but \(\xi :=B(\zeta )\not =1\), and so, by (8),
$$\begin{aligned} |A(\zeta )|&\le {} \sup _{\xi \not =1} \frac{|S''(\xi )|}{|S'(\xi )|^2}+ 2\sup _{\xi \not =1}\frac{|1-\xi |^2}{|S(\xi )|}\ C \\&\le {} C' \sup _{\xi \not =1} \frac{|1-\xi |^4}{|1-\xi |^4} +8C\le\infty .\end{aligned}$$
\(\square \)

Corollary 17

Let \(S_\mu \) be a singular inner function with finite spectrum \(\rho (S_\mu )\). Then \(S_\mu \in \mathfrak I_c\).

Proof

Since S is the universal covering map of \({\mathbb D}\setminus \{0\}\), each singular inner function \(S_\mu \) is written as \(S_\mu =S\circ v\) for some inner function v. Since \(\rho (S_\mu )\) is finite, v necessarily is a finite Blaschke product. (This can also be seen from [15, Proof of Theorem 2]). The assertion now follows from Proposition 16. \(\square \)

Note that the above result also follows in an elementary way from Proposition 12 and the fact that every such \(S_\mu \) is a finite product of powers of the atomic inner function S. We now consider left-compositions with finite Blaschke products.

Proposition 18

Let \(\Theta \) be a one-component inner function. Then each Frostman shift \((a-\Theta )/(1-\overline{a} \Theta )\in \mathfrak I_c,\) is too. Here \(a\in {\mathbb D}\).

Proof

Let \(\tau (z)=(a-z)/(1-\overline{a}z)\). Then \(\rho (\tau \circ \Theta )=\rho (\Theta )\). As stated above,
$$\begin{aligned} \liminf _{r\rightarrow 1}|\tau \circ \Theta (r\zeta )|<1 \end{aligned}$$
for every \(\zeta \in \rho (\tau \circ \Theta )\). Now
$$\begin{aligned} \tau (z)= \frac{1}{\overline{a}}+ \frac{|a|^2-1}{\overline{a}} \frac{1}{1-\overline{a} z}, \end{aligned}$$
from which we easily deduce the first and second derivatives. Using the formulas 9, we obtain
$$\begin{aligned} A:=\left| \frac{(\tau \circ \Theta )''}{[(\tau \circ \Theta )']^2}\right|\le \, {} C\frac{|1-\overline{a} \Theta |^4}{|1-\overline{a}\Theta |^3}+ C' |1-\overline{a} \Theta |^2 \; \frac{|\Theta ''|}{|\Theta '|^2}. \end{aligned}$$
Hence, the assumption \(\Theta \in \mathfrak I_c\) now yields (via Aleksandrov’s criterion, Theorem 15) that \(\sup _{\zeta \in \rho (\tau \circ \Theta )} A(\zeta )<\infty \). Thus, \(\tau \circ \Theta \in \mathfrak I_c.\) \(\square \)

Corollary 19

Given \(a\in {\mathbb D}\setminus \{0\}\) , the interpolating Blaschke products \((S-a)/(1-\overline{a} S)\) belong to \(\mathfrak I_c\).

This also follows from Corollary 9 by noticing that the a-points of S are located on a disk tangent at 1 and that the pseudohyperbolic distance between two consecutive ones is constant (see [20]). In the cited reference, it is also shown that the Frostman shift \((S-a)/(1-\overline{a} S)\) is an interpolating Blaschke product.

Corollary 20

Let B be a finite Blaschke product and \(\Theta \in \mathfrak I_c.\) Then \(B\circ \Theta \in \mathfrak I_c.\)

Proof

This is a combination of Propositions 12 and 18. \(\square \)

4 Inner functions not belonging to \(\mathfrak I_c\)

Here we present a class of Blaschke products that are not one-component inner functions. Recall that a Blaschke product b with zero-sequence \((z_n)\) is thin if
$$\begin{aligned} \lim _n\prod _{k\not =n} \rho (z_k,z_n)=\lim _{n\rightarrow 1} (1-|z_n|^2)|b'(z_n)|=1. \end{aligned}$$
It was shown by Tolokonnikov [24, Theorem 3] that b is thin if and only if
$$\begin{aligned} \lim _{|z|\rightarrow 1} (|b(z)|^2+(1-|z|^2)|b'(z)|)=1. \end{aligned}$$

Corollary 21

Thin Blaschke products are never one-component inner functions.

Proof

Let \(\varepsilon \in \;]0,1[\) be arbitrary close to 1. Choose \(\eta >0\) and \(\delta >0\) so close to 1 so that
$$\begin{aligned} \varepsilon<\eta ^2\, \text{ and }\, \displaystyle \eta < (1-\sqrt{1-\delta ^2})/ \delta . \end{aligned}$$
By deleting finitely many zeros, say \(z_1,\dots , z_N\) of b, we obtain a tail \(b_N\) such that \((1-|z_n|^2)|b_N'(z_n)|\ge \delta \) for every \(n>N\). Hence, by Theorem 5,
$$\begin{aligned} \{z \in {\mathbb D}: |b_N(z)|< \varepsilon \} \subseteq \{z \in {\mathbb D}: \rho (z, Z(b_N)) < \eta \} \end{aligned}$$
(10)
and the disks \({D_{\rho}}(z_n,\eta )\) are pairwise disjoint. This implies that the level set \(\{z \in {\mathbb D}: |b_N(z)| < \varepsilon \}\) is not connected. Now choose r so close to 1 that
$$\begin{aligned} p(z):=\prod _{n=1}^N \rho (z,z_n)\ge \varepsilon \end{aligned}$$
for every z with \(r\le |z|<1\). We show that the level set \(\{|b|<\varepsilon ^2\}\) is not connected. In fact, for some \(r\le |z|<1\) we have \(|b(z)|<\varepsilon ^2\), then
$$\begin{aligned} |b_N(z)|= \frac{|b(z)|}{|p(z)|}<\frac{\varepsilon ^2}{\varepsilon }=\varepsilon . \end{aligned}$$
Hence
$$\begin{aligned} \{z: r<|z|<1, |b(z)|<\varepsilon ^2\}\subseteq \{|b_N(z)|<\varepsilon \}\mathop \subseteq \limits _{}^{(3.1)} \bigcup _{n>N} {D_{\rho}}(z_n,\eta ). \end{aligned}$$
Since the disks \(D_\rho (z_n,\eta )\) are pairwise disjoint if \(n>N\), we are done with the tasks.\(\square \)

Corollary 22

No finite product B of thin interpolating Blaschke products belongs to \(\mathfrak I_c.\)

Proof

Let \(\varepsilon \in \;[0,1]\) be arbitrary close to 1. By Corollary 21, if \(b_j\), \((j=1,2)\), are two thin Blaschke products with zero-sequence \((z_n^{(j)})_n,\)
$$\begin{aligned} \Omega _{b_j}(\varepsilon )\subseteq \bigcup _{n=1}^\infty D_\rho (z^{(j)}_n,\eta ) \end{aligned}$$
for suitable \(\eta \), the disks \(D_\rho (z^{(j)}_n,\eta )\), being pairwise disjoint for n large. Since \(\lim _n\rho (z_n^{(j)}, z_{n+1}^{(j)})=1\), we see that a disk \(D_\rho (z^{(1)}_n,\eta )\) can meet at the most at one disk \(D_\rho (z^{(2)}_m,\eta )\) for n large. Hence
$$\begin{aligned} \Omega _{b_1b_2}(\varepsilon ^2)\subseteq \bigcup _{j=1}^2\bigcup _{n=1}^\infty D_\rho (z^{(j)}_n,\eta ), \end{aligned}$$
where the set on the right-hand side obviously is disconnected. The general case works via induction. \(\square \)

Remark 23

The conditions
$$\begin{aligned} \eta ^*:=\sup _{n\in { \mathbb N}}\rho (z_n, Z(b)\setminus \{z_n\})<1, \end{aligned}$$
(11)
or equivalently
$$\begin{aligned} D(z_n,\eta )\cap \bigcup _{m\not = n}D(z_m, \eta )\not =\emptyset \, \text{ for } \text{ some } \, \eta \in ]0,1[, \end{aligned}$$
(12)
are not sufficient to guarantee that the interpolating Blaschke product b is a one-component inner function.

Proof

Take \(z_{2n}=1-n^{-n}\) and \(z_{2n+1}=1-(n^{-n}+n^{-n})\). Then \((z_{2n})\) and \((z_{2n+1})\) are (thin) interpolating sequences by [16, Corollary 2.4]. Using \(a=n^{-n}\) and \(b=2a\), and the identity,
$$\begin{aligned} \rho (1-a,1-b)=\frac{|a-b|}{a+b-ab}, \end{aligned}$$
we conclude that
$$\begin{aligned} \rho (z_{2n}, z_{2n+1})= \frac{n^{-n}}{1-z_{2n} z_{2n+1}}\rightarrow 1/3, \end{aligned}$$
and so the union \((z_n)\) is an interpolating sequence satisfying (12). By Corollary 22, the Blaschke product formed with the zero-sequence \((z_n)\) is not in \(\mathfrak I_c.\) \(\square \)

Using the following theorem in [5], we can exclude a much larger class of Blaschke products from being one-component inner functions:

Theorem 24

(Berman) Let u be an inner function. Then, for every \(\varepsilon \in \;]0,1[,\) all the components of the level sets \(\{z\in { \mathbb C}: |u(z)|<\varepsilon \}\) have compact closures in \({\mathbb D}\) if and only if u is a Blaschke product and
$$\begin{aligned} \limsup _{r\rightarrow 1} |u(r\xi )|=1\, \textit{ for } \textit{ every }\, \xi \in \mathbb T. \end{aligned}$$
In particular this condition is satisfied by finite products of thin Blaschke products (see [17, Proposition 2]) as well as by the class of uniform Frostman Blaschke products whose zero sequence (z n ) satisfies
$$\begin{aligned} \sup _{\xi \in \mathbb T} \sum _{n=1}^\infty \frac{1-|z_n|^2}{|\xi -z_n|}<\infty . \end{aligned}$$
Note that this Frostman condition implies that the associated Blaschke product has radial limits of modulus one everywhere [9, p. 33]. As a byproduct of Theorem 6, we therefore obtain

Corollary 25

If b is a uniform Frostman Blaschke product with zeros \((z_n)\) clustering at a single point, then \(\limsup _\rho (z_n,z_{n+1})=1\).

Questions 25

To conclude, we would like to ask two questions and present three problems:
  1. (1)

    Can every inner function u whose boundary spectrum \(\rho (u)\) has measure zero, be multiplied by a one-component inner function into \(\mathfrak I_c\)?

     
  2. (2)

    Let \(S_\mu \) be a singular inner function with countable spectrum. Give a characterization of those measures \(\mu \) such that \(S_\mu \in \mathfrak I_c\). Do the same for singular continuous measures.

     
  3. (3)

    In terms of the zeros, give a characterization of those interpolating Blaschke products that belong to \(\mathfrak I_c\).

     
  4. (4)

    Does the Blaschke product B with zeros \(z_n=1-n^{-2}\) belong to \(\mathfrak I_c\)?

     
Footnotes
1

A shorter proof can be given using the advanced definition that a domain G in \({ \mathbb C}\) is simply connected if every curve in G is contractible in G, or equivalently, if for every Jordan curve J in G, the interior of J belongs to G. That depends, however, on the Jordan curve theorem.

 
2

This proof as well as two different ones, including the one mentioned in footnote 1, stem from the forthcoming book manuscript [22] of the second author together with R. Rupp.

 
3

This also follows form the inequalities \(1-|\sigma (\xi _n)|^2= \frac{(1-|a|^2)(1-|\xi _n|^2)}{|1-\overline{a} \xi _n|^2}\le \frac{1+|a|}{1-|a|} (1-|\xi _n|^2)\) and \(1-|\psi _n(a)|^2\le \frac{1+|a|}{1-|a|} (1-|w_n|^2)\), whenever \((w_n)\) is a Blaschke sequence and \(\psi _n(w_n)=\sigma (a)=0\).

 

Notes

Declarations

Acknowledgements

We thank Rudolf Rupp and Robert Burckel for their valuable comments concerning Lemma 2(1), the proof of which was originally developed for the upcoming monograph [22].

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, UNC
(2)
Département de Mathématiques et Institut Élie Cartan de Lorraine, Université de Lorraine

References

  1. Aleksandrov, A.B.: On embedding theorems for coinvariant subspaces of the shift operator. J. Math. Sci. 110, 2907–2929 (2002)MathSciNetView ArticleGoogle Scholar
  2. Aleman, A., Lyubarskii, Y., Malinnikova, E., Perfekt, K.-M.: Trace ideal criteria for embeddings and composition operators on model spaces. J. Funct. Anal. 270, 861–883 (2016)MathSciNetView ArticleMATHGoogle Scholar
  3. Baranov, A., Bessonov, R., Kapustin, V.: Symbols of truncated toeplitz operators. J. Funct. Anal. 261, 3437–3456 (2011)MathSciNetView ArticleMATHGoogle Scholar
  4. Belna, C.L., Obaid, S.A., Rung, D.C.: Geometric conditions for interpolation. Proc. Am. Math. Soc. 88, 469–475 (1983)MathSciNetView ArticleMATHGoogle Scholar
  5. Berman, R.: The level sets of the moduli of functions of bounded characteristic. Trans. Am. Math. Soc. 281, 725–744 (1984)MathSciNetView ArticleMATHGoogle Scholar
  6. Berman, R., Cohn, W.S.: Phragmén-Lindelöf theorems for subharmonic functions on the unit disk. Math. Scand. 62, 269–293 (1988)MathSciNetView ArticleMATHGoogle Scholar
  7. Bessonov, R.V.: Fredholmness and compactness of truncated Toeplitz and Hankel operators. Integral. Equ. Oper. Theory 82, 451–467 (2015)MathSciNetView ArticleMATHGoogle Scholar
  8. Cima, J., Wogen, W.: Isometric equivalence of isometries on \(H^p\). Proc. Am. Math. Soc. 144, 4887–4898 (2016)MathSciNetView ArticleMATHGoogle Scholar
  9. Collingwood, E., Lohwater, A.: Theory of cluster sets. Cambridge Univ Press, Cambridge (1966)View ArticleMATHGoogle Scholar
  10. Cohn, B.: Carleson measures for functions orthogonal to invariant subspaces. Pac. J. Math. 103, 347–364 (1982)MathSciNetView ArticleMATHGoogle Scholar
  11. Cowen, C., MacCluer, B.: Composition operators on spaces of analytic functions. CRC Press, New York (1995)MATHGoogle Scholar
  12. Gardiner, S.: Asymptotic maximum principles for subharmonic functions. Comp. Meth. Funct. Theory 8, 167–172 (2008)MathSciNetView ArticleMATHGoogle Scholar
  13. Garnett, J.B.: Bounded analytic functions. Academic Press, New York (1981)MATHGoogle Scholar
  14. Gerber, E.A., Weiss, M.L.: Interpolation, trivial and non-trivial homomorphisms in \(H^\infty \). J. Math. Soc. Japan 34, 173–185 (1982)MathSciNetView ArticleMATHGoogle Scholar
  15. Gorkin, P., Laroco, L., Mortini, R., Rupp, R.: Composition of inner functions. Results. Math. 25, 252–269 (1994)MathSciNetView ArticleMATHGoogle Scholar
  16. Gorkin, P., Mortini, R.: Asymptotic interpolating sequences in uniform algebras. J. Lond. Math. Soc. 67, 481–498 (2003)MathSciNetView ArticleMATHGoogle Scholar
  17. Gorkin, P., Mortini, R.: Cluster sets of interpolating Blaschke products. J. d’Analyse. Math. 96, 369–395 (2005)MathSciNetView ArticleMATHGoogle Scholar
  18. Hoffman, K.: Bounded analytic functions and Gleason parts. Ann. Math. 86, 74–111 (1967)MathSciNetView ArticleMATHGoogle Scholar
  19. Kerr-Lawson, A.: Some lemmas in interpolating Blaschke products and a correction. Can. J. Math. 21, 531–534 (1969)View ArticleMATHGoogle Scholar
  20. Mortini, R.: Commuting inner functions. J. Math. Anal. Appl. 209, 724–728 (1997)MathSciNetView ArticleMATHGoogle Scholar
  21. Mortini, R., Rupp, R.: On a family of pseudohyperbolic disks. Elem. Math. 70, 153–160 (2015)MathSciNetView ArticleMATHGoogle Scholar
  22. Mortini, R., Rupp, R.: An introduction to extension problems, Bézout equations and stable ranks in classical function algebras, accompanied by introductory chapters on point-set topology and function theory (in preparation).Google Scholar
  23. Shapiro, J.H.: Composition operators and classical function theory. Springer, New York (1993)View ArticleMATHGoogle Scholar
  24. Tolokonnikov, V.: Carleson–Blaschke products and Douglas algebras, Algebra i Analyz 3: 185–196 (Russian) and St. Petersburg Math. J. 3(1992), 881–892 (1991)Google Scholar
  25. Tse, Kam-Fook: Nontangential interpolating sequences and interpolation by normal functions. Proc. Am. Math. Soc. 29, 351–354 (1971)MathSciNetView ArticleMATHGoogle Scholar
  26. Tsuji, M.: Potential theory in modern function theory. Chelsea Pub Co, New Jersey (1975)MATHGoogle Scholar
  27. Weiss, M.L.: Some \(H^\infty \)-interpolating sequences and the behavior of certain of their Blaschke products. Trans. Am. Math. Soc. 209, 211–223 (1975)MathSciNetMATHGoogle Scholar
  28. Wortman, D.: Interpolating sequences on convex curves in the open unit disc. Proc. Am. Math. Soc. 48, 157–164 (1975)MathSciNetView ArticleMATHGoogle Scholar

Copyright

© The Author(s) 2017