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# One-component inner functions

- Joseph Cima
^{1}and - Raymond Mortini
^{2}Email authorView ORCID ID profile

**3**:2

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

© The Author(s) 2017

**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 functions
- Interpolating Blaschke products
- Connected components
- Level sets

## Mathematics Subject Classification

- Primary 30J10
- Secondary 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.

*S*is the atomic inner function, which is given by

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)
\(\Omega _0\)

*is a simply connected domain; that is,*\({ \mathbb C}\setminus \Omega _0\)*has no bounded components.*^{1} - (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 \(\{z\in {\mathbb D}: |f(z)|=\eta \}\not =\emptyset \).

*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,

*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 \(z_0\in \partial \Omega _0.\)

*H*. Consequently, \(|f|\le \eta \) on

*G*. By the local maximum principle, \(|f|<\eta \) on

*G*. Since \(\partial D\subseteq D\subseteq G\),

*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

*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

*z*to

*w*in \({\mathbb D}\) and

###
**Proposition 4**

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

###
*Proof*

*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

*n*,

*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,

*B*; a contradiction. \(\square \)

*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

*b*is bounded away from zero on \(\{z\in {\mathbb D}: \rho (z,Z(b))\ge \eta \}\).

###
**Theorem 5**

*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*

*If B is any interpolating Blaschke product with zeros*\(\{z_n:n\in { \mathbb N}\}\)

*such that*

*then*

- (1)
*The pseudohyperbolic disks*\(D_\rho (a,\eta )\)*for*\(a\in Z(B)\)*are pairwise disjoint.* - (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*

*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*

*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,

*n*implies that \(\bigcup _n D_\rho (z_n,\sigma )\) is connected. For the rest, just note that

###
**Corollary 7**

*Let B be a Blaschke product with increasing real zeros*\(x_n\)

*satisfying*

*Then*\(b\in \mathfrak I_c\).

###
*Proof*

*B*is interpolating if and only if

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:*

*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*

*n*-th iterate \(\tau _n\) of \(\tau \) are given by

*n*and \(\inf _{n\not =k}\rho (z_n,z_k)>0\). Now \((z_n)\) is a Blaschke sequence

^{3}([23, Ex. 6, p. 85]); in fact, use d’Alembert’s quotient criterion and observe that by the Denjoy–Wolff theorem,

*n*th Blaschke factor is positive at the origin).

*n*-th iterate \(\psi _n\) of \(\psi \) are given by

###
**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*

- (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). - (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.

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*

*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*

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 \(\Omega _{S(z^2)}(\eta ).\)
\(\square \)

###
**Theorem 15**

*Let*\(\Theta \)

*be an inner function. The following assertions are equivalent:*

- (1)
\(\Theta \in \mathfrak I_c.\)

- (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*

*B*on the boundary never vanishes (due to

*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:

###
**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*

###
**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*
\(B\circ \Theta \in \mathfrak I_c.\)

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

*b*with zero-sequence \((z_n)\) is

*thin*if

*b*is thin if and only if

###
**Corollary 21**

*Thin Blaschke products are never one-component inner functions.*

###
*Proof*

*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,

*r*so close to 1 that

*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

###
**Corollary 22**

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

###
*Proof*

*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

###
**Remark 23**

*b*is a one-component inner function.

###
*Proof*

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

###
**Theorem 24**

*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*

*z*

_{ n }) satisfies

###
**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**

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

- (4)
Does the Blaschke product

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

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.

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.

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 Access**This 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

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