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\begin{document}

\title{The convex hull of the interpolating
{B}laschke products}
\author{Michael D. O'Neill}



\address{ Department of Mathematics, UTEP\\El Paso, TX 79968}




\email{michael@math.utep.edu}


\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{claim}{Claim}
\newtheorem{theorem}{Theorem}
\newtheorem{Theorem}{Theorem 1}
\newtheorem{ex}{Example}
\newtheorem{rmk}{Remark}
\renewcommand{\theTheorem}{} 
\renewcommand{\thetheorem}{}
\newtheorem{prop}{Proposition}[section]
\def\pfclm{\noindent {\em Proof of claim}\quad}
\def\proof{\noindent{\em Proof.}\quad}
\def\pflm{\noindent {\em Proof of lemma.}\quad}
\def\pfthm{\noindent {\em Proof of theorem.}\quad}
\def\rse#1{\mbox{$^{\raisebox{.3ex}{$\scriptstyle{#1}$}}$}}

\maketitle
\openup 3\jot
\large
\section{Introduction and Notations } 
In the sequel we prove that if a Blaschke product $B$ is continuous in the closed unit disk
except on a closed set $E\subset \Bbb T$ of measure zero, then $B$ is contained
in $9K$, where $K$ denotes the closed convex hull of the interpolating Blaschke
products. Moreover, we show that a generic Blaschke product is contained in
$27K$. By the well known theorem of Marshall, this implies that the
unit ball of $H^\infty$ is contained in $27K$. The proofs employ a technical result, given in the first section, 
which may be of some independent interest. 
The results in the paper improve earlier work in
\cite{M-S},\cite{G-N} and \cite{Oy}. We refer to these papers and to the book 
\cite{G} for further         
background on the questions treated here. 

This paper is based on  part of the author's UCLA dissertation
completed in
 1995
under the direction of John Garnett. I thank him for all his help
and encouragement. My thanks also go to the referee and the editor for their help in
clarifying the presentation. 

\begin{description}
 \item[$\Bbb D$]          ----- Open unit disk,  $\qquad\Bbb D \equiv
                                                  \{z\in \Bbb C : |z| <1\}$ 
\\
 \item[$\Bbb T$]          ----- Unit circle,     $\qquad \Bbb T \equiv
                                                  \partial \Bbb D$
\\
 \item[$H\rse{\infty}$]   ----- The space of bounded analytic
                                        functions in $\Bbb D$.
\\
 \item[$L\rse{\infty}$]   ----- The space of essentially bounded
                                 functions on $\Bbb T$.
\\
 \item[$P_z(w)$]          ----- The Poisson kernel in $\Bbb D$,
                         $\qquad P_z(w)= \frac{1-|z|^2}{|1-\overline w z|^2}.$
\\
 \item[$\rho(z,w)$]       ----- The ``pseudo hyperbolic distance''
between $z$ and $w$ in $\Bbb D$,$$ \rho(z,w) \equiv \bigl
|\frac{z-w}{1- \overline{w}z} \bigr|$$ 
 \item[$d(z,w)$]      ----- The hyperbolic distance between $z$ and
$w$ in $\Bbb D$. $$ d(z,w) \equiv \log{\frac{1+ \rho(z,w)}{1-
\rho(z,w)}}$$
\item[$d(X,Y)$]       ----- For $X\subset \Bbb D, Y \subset \Bbb
D$,
$$
\d(X,Y) \equiv \inf\{d(x,y):x\in X,y\in Y\}
$$
\item[$A_{\delta}^{\epsilon}$]  ----- An annulus of thickness
$2\epsilon$ about the circle $\{|z| = \delta \}$
$$
A_{\delta}^{\epsilon} \equiv \{z\in \Bbb D : \delta - \epsilon \le |z|
\le
\delta + \epsilon\}$$
\item[$I$]             ----- A half open subinterval of $\Bbb T$
$$ I = \{e^{i\theta} \in \Bbb T: \theta \sb 1 \le \theta < \theta \sb
2 \} \qquad \text{for some} \quad 0\le \theta \sb 1 < \theta \sb 2 \le
2\pi.$$
\item[$|I|$]   ----- $|I| = \theta \sb 2 - \theta \sb 1 $, the arc
length of $I$.
\\
\item[$Q=Q(I)$]    ----- A Carleson \lq \lq square\rq\rq \ in $\overline
{\Bbb D}$.
$$Q\equiv \{re^{i\theta}: e^{i\theta}\in I, 1- |I| \le r \le 1 \}
\qquad \text{for some} \quad I \subset \Bbb T$$
\item[$\ell (Q(I))$] ----- $\ell(Q)\equiv |I| $
\\
\item[$\beta Q$] ----- For $\beta \in \Bbb R ^+ $,
$$
\beta Q \equiv \{re^{i\theta}: \frac{\theta \sb 1 + \theta \sb 2}2
- \beta(\frac{\theta \sb 2 - \theta \sb 1}2) \le \theta <
 \frac{\theta \sb 1 + \theta \sb 2}2+
 \beta(\frac{\theta \sb 2 - \theta \sb 1}2), \quad 1- \beta (\theta
\sb 2 - \theta \sb 1) \le r \le 1 \}
$$                
\\
\item[$T(Q)$] ----- $T(Q) \equiv Q\cap \{ z\in \Bbb D : |z| < 1-
\frac{\ell (Q)}2 \}$
\\
\item[$u\sb a$] ----- For an inner function $u$ and $a\in \Bbb D$ the
inner function $u\sb a$ is defined by
$$
u\sb a = \frac{u-a}{1-\overline{a} u}.
$$
\item[$B|\sb U$] ----- If $B$ is a given Blaschke product with 
zeros $\{z\sb {\nu}\}$  and $U$ 
                       a subset of the open unit disk, then
$$
B|\sb U \equiv \prod \limits \sb {z\sb {\nu} \in U} \frac {-\overline{ z \sb \nu}}
{|z\sb \nu|}\cdot \left(\frac {z-z\sb \nu}{1-\overline{ z\sb \nu} z
}\right) 
$$
\item[$\mathcal I$] ----- The set of interpolating Blaschke products.
\\
\item[$K$] ----- The closed convex hull of $\mathcal I$.
\\
\item[$\mathcal F$] ----- The set of finite products of interpolating
Blaschke
products.
\end{description}
\section{Preliminaries}
This section includes some background on  the
results presented in the paper and on two interesting open questions.
Several results recorded here will be referred to later.

A holomorphic function $u\in H\rse{\infty}(\Bbb D)$ is called an
inner function if its radial limits (which exist a.e. on $\Bbb T$ by
Fatou's theorem) satisfy $|u(e^{i\theta})| =1
$ a.e on $ \Bbb T$.

An example of an inner function is 
$$
u(z) =e^{i \alpha} z^m \prod \limits \sb {\nu=1} ^{\infty} \frac {-\overline{ z \sb \nu}}
{|z\sb \nu|}\cdot \left(\frac {z-z\sb \nu}{1-\overline{ z\sb \nu} z
}\right) 
$$
where the convergence of the infinite product is
assured by requiring that $$\sum \limits \sb {\nu} (1-|z\sb {\nu}|) <
+\infty,$$ where each zero $z_{\nu}$ is counted with its multiplicity. 
An inner function of this type is called a Blaschke product.

There are non-constant inner functions with no zeros in $\Bbb D$
called singular functions. They are
$$
u(z) = \exp\left(-\int \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\lambda (e^{i\theta})
\right)
$$
where $d\lambda$ denotes a positive measure on $\Bbb T$ which is
singular with respect to the Lebesgue measure.
Now, by the canonical factorization theorem for $H\rse{\infty}$, the
only
other examples of inner functions in $\Bbb D$ are constant multiples
of products of the
above two types. The following theorem
shows that the Blaschke products are uniformly dense in the inner functions. 
See \cite{G} for details.
\begin{thm}[Frostman] 
Let $v(z)$ be a non-constant inner function on the unit disc. Then for all
$\zeta$ with $|\zeta|<1$, except possibly for a set of logarithmic capacity 
zero, the function
$$ v\sb \zeta (z) = \frac{v(z)-\zeta}{1-\overline{\zeta}v(z)}
$$
is a Blaschke product. 
\end{thm}


This paper is concerned with those  Blaschke
products
which have simple zeros lying in an interpolating sequence, 
that is, a sequence of points in  $\Bbb D$ such that each
interpolation problem
$$
f(z\sb {\nu}) = w\sb {\nu} \qquad \nu = 1,2,\dots \quad \{w\sb
{\nu}\}\in \ell\rse{\infty}
$$
has a solution $f\in H\rse{\infty}$. The basic facts on interpolating
sequences
are contained in the following theorem of Carleson.
\begin{thm}[Carleson] If $\{z\sb j \} $ is a sequence in the unit disc,
then the following conditions are equivalent:

\begin{enumerate}
\item The sequence is an interpolating sequence.
\item There is  a $\delta >0$ such that
$$
\prod \limits \sb {j,j\ne k}\left |\frac {z\sb k -z\sb j}{1-\bar z \sb j z\sb k
 }
\right| \ge \delta, \qquad k=1,2,\dots
$$
\item The points $z \sb j $ are separated,
$$ \left |\frac{z\sb j - z\sb k }{1-\bar z\sb j z \sb k}
\right| \ge a>0, \qquad j\ne k,
$$ 
and there is a constant C such that for every \lq\lq square" $Q$
%$$ Q=\{re^{i\theta} : \theta \sb 0<\theta< \theta \sb 0 + \ell (Q) ,
% 1-\ell(Q) <r<1 \}
%$$
we have
$$ \sum \limits \sb {z\sb j \in Q } 1-|z\sb j| \le C\ell (Q) .
$$
\end{enumerate}
\end{thm}

A Blaschke product with simple zeros on an interpolating sequence 
is called an interpolating Blaschke
product. We write $\mathcal I$ for the set of interpolationg Blaschke
products, and $\mathcal F$ for the set of finite products of 
interpolationg Blaschke products.
 It is known, for example, that every uniformly
closed algebra between $H\rse{\infty}$ and $L\rse{\infty}$ is
generated
by $H\rse{\infty}$ and the complex conjugates of a subset of $\mathcal
I$.
See \cite{C} and \cite{M}.
The constructive proof of the Douglas-Rudin theorem due to
Jones,\cite{J1},
shows that any unimodular function in $L\rse{\infty}(\Bbb T)$ may be
uniformly
approximated by a ratio of products in $\mathcal I$.
 
 The following two problems are posed in (\cite{G}, p.430)    
\begin{description}
\item[Problem 1]  Do the interpolating Blaschke products generate 
$H\rse{\infty}$ as a uniform algebra?
\item[Problem 2]  Can every Blaschke product be uniformly approximated
by interpolating Blaschke products?

\end{description}
It is known that the Blaschke products do generate $H\rse{\infty}$ and
in fact that the closed convex hull of the Blaschke products is the
unit ball of $H\rse{\infty}$,
see \cite{Mar}. 

Problem 1 was recently settled in the affirmative in \cite{G-N},
following
the work in \cite{M-S} in which any Blaschke product with zeros which
accumulate only at a set of measure zero on $\Bbb T$ was shown to be
in the linear hull of $\mathcal I$. The idea in \cite{G-N} is to
carefully
factor a general Blaschke product into finitely many subproducts, to
each
of which a version  of the argument of \cite{M-S} may be applied.
This factorization technique solves Problem 1 because of the following
lemma from \cite{M-S}.

\begin{lemma}[Marshall, Stray] Given a Blaschke product $ B$ which is
the product of finitely many interpolating Blaschke products and 
given $\epsilon >0$
 there exists an interpolating Blaschke product $B^* $ such that

 $$
\sup \limits \sb {z\in \Bbb D } |B(z)-B^*(z)| <\epsilon .
$$
\end{lemma}

The lemma shows that we may as well pose problems 1 and 2 for the
class
$\mathcal F$ instead of $\mathcal I$. A characterization of $\mathcal F$
in terms of zero sequences is given in \cite{G-N}.

\begin{lemma}[Garnett, Nicolau] Let $B$ be a Blaschke product and
let $\{z\sb \nu \} $ be its zeros, counted with their multiplicities.
Then the following are equivalent:
\begin{enumerate}
\item $B=B\sb 1 \dots B\sb N $ ,  and each $B\sb j $ is an interpolating
Blaschke product.
\item There is a constant C such that for every \lq\lq square" $Q$
%$$ Q=\{re^{i\theta} : \theta \sb 0<\theta< \theta \sb 0 + \ell (Q) ,
% 1-\ell(Q) <r<1 \}
%$$
we have
$$ \sum \limits \sb {z\sb j \in Q } 1-|z\sb j| \le C\ell (Q) .
$$
\item There exist positive constants $\rho \sb 0, \delta \sb 0 $ such that
for each $z\sb \nu $ there is a $w\sb \nu $ with
$$
\left|\frac{z\sb {\nu} - w\sb {\nu}}{1- \overline{w\sb {\nu}}z\sb
{\nu}}\right|
\le \rho \sb 0
$$
and
$$
(1-|w\sb \nu|^2)|B' (w\sb \nu )| \ge \delta \sb 0.
$$
\end{enumerate}
\end{lemma}

Problem 2 remains open, as does the following, intermediate between
Problem 1 and 2.
\begin{description}
\item[Problem 1.5] Is the unit ball of $H\rse{\infty}$ the closed
convex hull of $\mathcal I$?
\end{description}

In \cite{Oy}, K.{\O}yma was able to show that $K$, the convex hull of 
$\mathcal I$, has non-empty interior in $H\rse{\infty}.$ Here we
improve this result to show that $K$ contains all $H\rse{\infty}$
functions
with norm not greater than $\frac{1}{27}$, by expanding on some 
ideas from 
the paper \cite{B}. See also the paper \cite{N} for some similar
arguments.

In section 4
we will need the following lemma and  remarks .
They are from the exposition in \cite{G} of Hoffman's theory
of the maximal ideal space of $H\rse{\infty}$. 
\begin{lemma}[Hoffman]
 Let $B(z)$ be an interpolating Blaschke product
 with zeros $S=\{z\sb n\}$,
 and suppose
$$
\inf \limits \sb n (1-|z\sb n|^2)|B'(z\sb n)| \ge\delta>0.
$$
There exist $\lambda=\lambda(\delta),0<\lambda <1$, and $r=r(\delta),0<r<1$,
satisfying
$$
\lim \limits \sb{\delta \to 1} \lambda(\delta)=1
$$
$$
\lim \limits \sb{\delta \to 1} r(\delta)=1
$$
and having the following properties: The set
 $B^{-1}(\Delta(0,r))=\{z:|B(z)|<r\}$
is the union of pairwise disjoint domains $V\sb n$, $z\sb n \in V\sb n$, and
$$
V\sb n \subset \{z:\left|\frac{z-z\sb n}{1-\overline{z\sb n}z}\right|<\lambda\}.
$$
$B(z)$ maps each domain $V\sb n$ univalently onto $\Delta(0,r)=\{w:|w|<r\}$.
If $|w|<r$, then
$$
B\sb w(z)= \frac{B(z)-w}{1-\overline{w}B(z)}
$$
is an interpolating Blaschke product having one zero in each $V\sb n$.
\end{lemma}
\begin{rmk}\emph{ In the proof of lemma 2.3 the numbers $\lambda$ and $r$ are chosen according
to}
\begin{enumerate}
\item
$
\displaystyle\lambda<\frac{2\lambda}{1+\lambda^2}<\delta
$
\item
$
\displaystyle\lambda=\lambda(\delta)\to 1 \text{ and } \frac{\delta-\lambda}{1-\lambda\delta}
\to 1 \qquad  \mbox{as }\delta \to 1
$
\item
$
\displaystyle r(\delta)= \frac{\delta-\lambda}{1-\lambda\delta}\lambda \to 1.
$
\end{enumerate}
\end{rmk}

\begin{rmk}\emph{
Let $u\in \mathcal F$ have the zero sequence $\{z\sb n\}$
 where multiple zeros are repeated and write
$$
u=\prod \limits \sb {i=1} ^N B\sb i \qquad B\sb i
 \in \mathcal I,\quad 1\le i\le N.
$$
then if $\lambda >0$ is given and $\epsilon(\lambda) >0$ is
sufficiently small,
we have by lemma 2.3
$$
\{z\in \mathbb D : |u(z)|\le \epsilon \} \subset
\bigcup \limits \sb {i=1} ^N \{z\in \mathbb D : |B\sb i(z)| \le
\epsilon^{\frac1N}\} \subset \bigcup \limits \sb n \{z:\rho(z,z\sb n) < \lambda \}.
$$
If $\lambda >0$ is sufficiently small then
by the density condition on the set $\{z\sb n\}$ there is an $M>0$
such that any connected component of $\bigcup \limits \sb n
\{z:\rho(z,z\sb n) < \lambda \}$ is the union of at most $M$ of the
disks $\{z:\rho(z,z\sb n) < \lambda \}$. By the argument principle, 
$u\sb a \in \mathcal F$ whenever $|a| <\epsilon(\lambda)$. This shows
that if $v$ is any inner function, then the set 
$\left\{a \in \Bbb D: v\sb a \in
 \cal F\right\}$
is open.} 
\end{rmk}
 The last (well known) lemma of this
section
 gives us information on the modulus of a Blaschke product in
terms of its zero sequence. It's proof is an exercise with the
identity of Lagrange:
$$
\frac{(1-|z|^2)(1-|w|^2)}{|1-\overline{w}z|^2}=1-\left|\frac{z-w}
{1-\overline{w}z} \right|^2 \qquad \text{for all}\quad z,w \in  \mathbb D
$$

\begin{lemma}
Let $0<a<1$ and let $B$ be a Blaschke product in the disk with 
zero sequence $\{z\sb {\mu}\}$ where multiple zeros are repeated in
the
sequence. Suppose that
$$
\left| \frac{z-z\sb {\mu}}{1-\overline{z\sb{\mu}}z}\right| >a >0
\qquad \text{for each}\quad \mu.
$$
Then
$$
\frac12 \sum \limits \sb {\mu}
\frac{(1-|z|^2)(1-|z\sb {\mu}|^2)}{|1-\overline{z\sb {\mu}}z|^2}
\le \log{\frac1{|B(z)|}}
\le
\frac12 C(a)\sum \limits \sb {\mu}
\frac{(1-|z|^2)(1-|z\sb {\mu}|^2)}{|1-\overline{z\sb {\mu}}z|^2}.
$$
The left hand inequality holds without any condition except $0<a<1$ and
  we have $C(a)\to 1 $ as  $a \to 1$
\end{lemma}
\proof See (\cite{G}, p. 288-289).
%Writing
%$$
%\log{\frac1{|B(z)|}} = - \frac12 \sum \limits \sb{\mu} 
%\log\left(1-\left(1- \left|\frac{z-z\sb {\mu}}{1- \overline{z\sb
%{\mu}}z}\right|^2 \right)\right)
%$$
%the proof is completed with an application of the Lagrange identity
%and
%the inequalities
%$$
%x \le -\log(1-x)
%$$ 
%valid for $x<1$ and
%$$
%-\log(1-x) \le C(a)x
%$$ valid for $0 \le x < 1-a^2$.
%
\section{ An approximation theorem}  

In this section we use ideas from Bishop's characterization of the
zero sets of Blaschke products in the little Bloch space to prove 
a technical theorem. See \cite {B}. The theorem will be used in section 4 to factor
Blaschke products in a procedure similar to that used in \cite{G-N}.
If $u$ is any inner function with the canonical factorization
$$
u(z) = \prod \limits \sb {z\sb \nu}\frac {-\overline{ z \sb \nu}}
{|z\sb \nu|}\cdot \left(\frac {z-z\sb \nu}{1-\overline{ z\sb \nu} z }\right) 
\exp\left(-\int \frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\lambda (e^{i\theta})
\right) ,
$$
 we associate  with $u$ a measure $\mu \sb u$ defined by
$$
\mu \sb u \equiv \sum (1-|z\sb \nu|) \delta \sb {z\sb \nu} +d\lambda .
$$
Here $\lambda$ is the non-negative singular measure occuring in the singular
factor of $u$ and $\{z\sb \nu\}$ is the zero set of $u$. Multiple
zeros are repeated in the sequence according to their multiplicity.

\begin{thm}Let $u$ be a given inner function and let $1>\delta >0$ be
fixed. Let $E\sb {\delta}=\{z\in \mathbb D: |u(z)|= \delta\}.$
 The following two statements
are equivalent.
\begin{enumerate}
\item There exists an $\epsilon>0$ such that
$$
u\sb a\in \cal F, \quad \text{for each}\quad a \in A\sb{\delta}^{\epsilon}.
$$
\item There are
$ M>0 $ and $ \eta>0$ such that for each $z\in E\sb \delta$,
 there exists a square $Q$ with the following two properties:
 \begin{enumerate}
 \item
 $d(z,T(Q))<M$,
 \item
 $ \frac{\mu \sb u(Q)}{\ell(Q)}\notin[\frac1{2\pi}(\log \frac1{\delta} -\eta),
 \frac1{2\pi}(\log \frac1{\delta} + \eta)]. $
 \end{enumerate}
\end{enumerate}
\end{thm}

The theorem and lemma 2.1 imply that if we can factor a given finite
Blaschke
product into $o(\frac1{\delta})$ factors, each of which
satisfies condition 2 with $M$ and $\eta$ independent of $\delta$
then we can solve Problem 2.  This fact had been noticed independently by
K. {\O}yma and communicated to the author in June 1996. 

Before beginning the proof of the theorem we record, in the following lemma,
some remarks from \cite{B}.
\begin{lemma}
There exists a universal constant $\alpha$ with $0<\alpha<1$ such that for any 
inner
 function $u$, if $\mu \sb u (16Q)=0$ for some square $Q\subset \overline
{\Bbb D}$ with $\ell(Q)<\frac1{32}$, then for any $e^{i\theta} \in Q$ and for
$w\sb n=(1-2^{-n}\ell(Q))e^{i\theta}$, we have
$$
|u(w\sb n)|\ge|u(w\sb {n-1})|^{\alpha} \qquad \text{for each}\quad n\ge 1.
$$
\end{lemma}
 See (\cite{B}, p. 215 - 216) for a proof of the lemma. 
% Examining the form of a general inner function, we see that it 
% suffices to show that for $z\in\overline {\Bbb D}\setminus 16Q$ we have
%$$
%(*)\qquad \frac{1-|w\sb n|^2}{|1-\overline{w\sb n}z|^2}\le \alpha
%\frac{1-|w\sb {n-1}|^2}{|1-\overline{w\sb {n-1}}z|^2}
%$$
%and 
%$$
%(**)\qquad \left|\frac{z-w\sb n}{1-\overline{w\sb n}z}\right|
%\ge \left|\frac{z-w\sb {n-1}}{1-\overline{w\sb {n-1}}z}\right|^{\alpha}.
%$$
%With $|w\sb 0|\ge \frac{31}{32}$ and $z\in\overline {\Bbb D}\setminus 16Q$
%we have
%$$
%\left|\frac{1-\overline{w\sb {n-1}}z}{1-\overline{w\sb n}z}\right|
%\le 1+\frac{|w\sb n- w\sb {n-1}|}{|1-\overline{w\sb n}z|} < \frac{17}{15}
%$$
%so that
%$$
%\frac{1-|w\sb n|^2}{1-|w\sb {n-1}|^2}\left|\frac{1-\overline{w\sb {n-1}}z}{1-\%overline{w\sb n}z}\right|^2
%< \frac12 \cdot   \frac{1+|w\sb n|}{1+|w\sb {n-1}|}\cdot
%\left(\frac{17}{15}\right)^2 <
%\alpha <1
%$$
%and this is $(*)$.
%Using the  Lagrange identity we also have
%$$
%1-\left|\frac{z-w\sb n}{1-\overline{w\sb n}z}\right|^2  \le
%\alpha\left(1-\left|\frac{z-w\sb {n-1}}{1-\overline{w\sb {n-1}}z}\right|^2
%\right).
%$$
%Inequality $(**)$ now follows from the inequality
%$$
%x^{\alpha}\le 1-\alpha(1-x)
%$$
%valid for $0<x<1$ and $0<\alpha<1$. 
\\
\\
\pfthm

For any $z\in \mathbb D$ let $Q\sb z$ be a square which
 has $z$ at the center of its top edge
and let  
$$
S(z,n)\equiv \left\{z' \in Q\sb
z : 2^{-(2n)}(1-|z|) \le( 1-|z'|) \le 2^{-n}(1-|z|)\right \}.
$$
Suppose that condition 2 of the theorem fails for the given inner
function $u$. This means that if we are given $\eta>0$ and $M>0$, we
can find some
  $ z= z(\eta, M)\in E\sb {\delta} $ such that
 all squares $Q$ with $d(z,T(Q))<M$ satisfy
$$ 
\frac{\mu \sb u(Q)}{\ell(Q)}\in[\frac1{2\pi}(\log \frac1{\delta} -\eta),
\frac1{2\pi}(\log \frac1{\delta} + \eta)]. 
$$

With $\eta$ and $\delta$ fixed, let $N$ be a positive integer
and let $z= z(\eta,M)$.
Factor $u$
as $u= u\sb 1 u\sb 2$ so that $\mu \sb {u\sb 2}$ is supported in 
$\overline{\Bbb D}/NQ\sb z$. By lemma 2.4, if $N= N(\delta)>0$ is sufficiently large
then
 $|u\sb 2(w)| > \frac{\delta}2$ for all $w$ on
the 
top edge of $Q\sb z$. We further require that
$$
(\frac{\delta}2)^{\alpha^N} > e^{-\eta}
$$
where $\alpha $ is the constant from lemma 3.1
and that
$$
\int \limits \sb {\mathbb T \setminus NQ\sb z}
P\sb {z'}(e^{i\theta})\,d\theta \le \frac {\eta}{10\log{\frac 1{\delta}}}
\qquad \text{for each} \quad z' \in S(z,N).
$$
Let $\mathcal C_N$
be the collection of $N 2^{2N}$ disjoint Carleson squares
 contained in $NQ\sb z$, with
$$
\ell (Q\sb j) = 2^{-2N} \ell(Q\sb z) \qquad \text{for each} \quad Q\sb j \in
\mathcal C_N.
$$
Note that any square $S$ whose top half contains a zero has
$\frac{\mu(S)}{\ell(S)}$
differing by at least $\frac12$ from some square of exactly half the
size
contained inside of it.
So if $M$ is large enough and $\eta< \frac12$ then $\mu_{u\sb 1}$ is
supported
in $\bigcup \limits \sb {\cal C_N} Q\sb
j$.

We have
\begin{eqnarray}
\biggl|-\log{|u(z')|}& -& 2\pi \frac{\mu \sb u (Q\sb z)}{\ell (Q\sb z)}
\biggr| \le 
\biggl|-\log{|u(z')|} +\log{|u\sb 1 (z')| }
\biggr|  \nonumber 
\\{}\nonumber\\&+& \biggl|\log{\frac1{|u\sb 1 (z')|}} - \int \limits \sb {NQ\sb z} 
\frac{1-|z'|^2}{|1-\overline{w}z'|^2}\,d\mu\sb u (w) \biggr| 
\nonumber
\\{}\nonumber\\&+&\biggl|\int \limits \sb {NQ\sb z} 
\frac{1-|z'|^2}{|1-\overline{w}z'|^2}\,d\mu\sb u (w) -
\sum\limits \sb {Q\sb j \in \cal C_N}\frac
{\mu\sb u(Q\sb z)}{\ell(Q\sb z)}\int \limits \sb {Q\sb j \cap \Bbb T}
P\sb {z'}(e^{i\theta})\,d\theta \biggr|
\nonumber
\\{}\nonumber\\ &+&\biggl|\sum\limits \sb {Q\sb j \in \cal C_N}\frac
{\mu\sb u(Q\sb z)}{\ell(Q\sb z)}\int \limits \sb {Q\sb j \cap \Bbb T}
P\sb {z'}(e^{i\theta})\,d\theta
-
2\pi \frac{\mu \sb u (Q\sb z)}{\ell (Q\sb z)}
\biggr|
\nonumber
\end{eqnarray}
for any $z'\in S(z,N)$.
The first and fourth terms on the right are each smaller than
$\eta$
by our choice of $N$. By application of lemma 2.4 and properties of the
Poisson
kernel respectively, the second and
third terms are each smaller than $\eta$ if $M$ is
sufficiently
large. This shows that $||u(z')|- \delta
| < 4\eta  $ for any $z'\in S(z,N)$. As $N$ can be arbitrarily large,
Schwarz's lemma and remark 2 after lemma 2.3 then show that condition
(1) of the theorem fails.


For the remaining half of the proof, let $Q\sb z$ be as
before,
the square with $z$ at the center of its top edge.
Let $I=Q\sb z \cap \Bbb T$ denote the base of $Q\sb z$ and let
$S\sb m$ denote the union of $2^m$ disjoint Carleson squares of length    
$r_m\equiv 2^{-m}\ell(Q\sb z)$ contained in $Q\sb z$.
Suppose now that for any given $\tau>0$ and  integer $n>0$,
there exists a point $z\sb 0(\tau,n)\in \Bbb D$ such that
 $|u(z\sb 0)|=\delta$,
but $|u(z\sb 0)-u(w)|<\tau$ for all $w\in  2^nQ\sb {z\sb 0}\cap \{\xi
\in \mathbb D:|\xi|\le1-2^{-2n}\ell(Q\sb {z\sb 0})\}$.
By lemma 2.2 and Schwarz's lemma, this is true for any inner function  $u$ for  which
condition (1) of the theorem fails.
 We will
show that given $M>0$ and $\eta>0$, if $n$ is large enough and $\tau$ small
enough, then all squares $Q$ with $d(z\sb 0,T(Q))<M$ have
$$
 \frac{\mu \sb u(Q)}{\ell(Q)}\in[\frac1{2\pi}(\log \frac1{\delta} -\eta),
\frac1{2\pi}(\log \frac1{\delta} + \eta)].
$$
That will finish the proof of the theorem.
 Choose a point $z$ with $d(z,z\sb 0)<M$.
 With sufficiently large $n$ and small $\tau$, we  have
  $$
\left|\int \limits \sb {\overline{\Bbb D}}P\sb {\xi}(w)\,d\mu\sb u (w)
+\log {|u(z\sb 0)|}\right| < \frac {\eta}4 \quad
 \text{for each}\quad \xi\in Q\sb z/S\sb n.
$$
In fact,
$$
\int \limits \sb {\overline{\Bbb D}}P\sb {\xi}(w)\,d\mu\sb u (w)
= \sum \limits _{z\sb {\nu}}\frac{(1-|\xi|^2)(1-|z\sb
{\nu}|)}{|1-\overline{z\sb{\nu}}\xi|^2} + \int \limits \sb {\Bbb
T}\frac{(1-|\xi|^2)}{|1-\overline{z\sb{\nu}}\xi|^2}\, d\lambda .
$$
and all values of $|u(\xi)|$ are very near $\delta$ for a very large 
hyperbolic distance around $z\sb 0$. So by lemma 3.1, the zeros $z\sb
{\nu}$
which are outside of $2^n Q\sb {z\sb 0}$ make a small contribution to
the sum, and lemma 2.4 implies
$$
\int \limits \sb {\overline{\Bbb D}}P\sb {\xi}(w)\,d\mu\sb u (w)
\sim \log{\frac1{|u(\xi)|}}.
$$


Let $\hat I = (1+2^{-\frac{n}2})I$ and let $c= \frac{\eta}{8|\log \delta|}$.
For sufficiently large $n$, we have 
$$
\int \limits \sb {\hat I} P\sb {(1-r_n)e^{i\theta}}(w)\, \frac{d\theta}{2\pi}
=
\frac{1-(1-r_n)^2}{1-|(1-r_n)w|^2} \int \limits \sb {\hat I}\frac{1-|(1-r_n)w|^2} 
 {|1-e^{-i\theta}(1-r_n)w|^2}\,\frac{d\theta}{2\pi}
 \ge 1-c
$$
for all $w\in S_{2n}$  because the second integral is the
harmonic measure at $(1-r\sb n)w$ of the arc $\hat I$. Therefore,
 $$
\frac{\mu \sb u(Q\sb z)}{\ell(Q\sb z)}=\frac{\mu \sb u(S\sb {2n})}{\ell(Q\sb z)}
\le \frac1{(1-c)\ell(Q\sb z)}\int \limits \sb {S\sb {2n}}
\left \{ \int \limits \sb {\hat I} P\sb {(1-r_n)e^{i\theta}}(w)\, \frac{d\theta}
{2\pi}\right \}\,d\mu\sb u (w).
$$
$$
\le (1+2c) \frac{|\hat I|}{2\pi \ell(Q\sb z)}\int \limits
 \sb {S\sb {2n}} P\sb {(1-r_n)e^{i\theta^*}}(w)\,d\mu\sb u(w)
$$
(for the appropriate $\theta^*$ with $e^{i\theta^*}\in \hat I$ )
$$
\le \frac{1+4c}{2\pi}\int \limits \sb {\overline {\Bbb D}}
  P\sb {(1-r_n)e^{i\theta^*}}(w)\,d\mu\sb u(w) 
$$
(for sufficiently large $n$)
$$
\le \frac{1+4c}{2\pi} (-\log{|u(z\sb 0)|} + \frac{\eta}4 )
$$
$$
\le -\frac1{2\pi}\log{|u(z\sb 0)|} +\eta.
$$


Now let $\tilde I = (1-2^{-\frac{n}2})I$ and let $r\sb n =
2^{-n}\ell(Q\sb z)$. Let $N$ be a positive
integer to be chosen below and for any $e^{i\theta} \in \tilde I$
let $Q\sb {e^{i\theta}}$ be the
 square with $\ell(Q\sb {e^{i\theta}}) =Nr_n$ and base
 centered at $e^{i\theta}$. By lemma 2.4 and lemma 3.1, if
$n$ is large enough and $\tau$ small enough we
 may choose a large $N<2^{\frac{n}2}$ 
such that 
$$
\left |\,\int \limits \sb {\overline {\Bbb D}\setminus Q\sb {e^{i\theta}}} 
P\sb {(1-r_n)e^{i\theta}}(w)\,d\mu\sb u(w)\right | < \frac{\eta}8.
$$
We therefore have for all $e^{i\theta}\in \tilde I$
$$
\left|\,\int \limits \sb {S\sb n} P\sb {(1-r_n)e^{i\theta}}(w)\,d\mu\sb u(w)
-\int \limits \sb {\overline {\Bbb D}} 
P\sb {(1-r_n)e^{i\theta}}(w)\,d\mu\sb u(w)\right|<\frac{\eta}4
$$
and therefore
$$
\frac{\mu \sb u(Q\sb z)}{\ell(Q\sb z)}=\frac{\mu \sb u(S\sb n)}{\ell(Q\sb z)}
\ge\frac1{\ell(Q\sb z)}
\int \limits \sb {S\sb n}
\left \{ \int \limits \sb {\tilde I} P\sb {(1-r_n)e^{i\theta}}(w)\, \frac{d\theta}
{2\pi}\right \}\,d\mu\sb u (w)
$$
$$
=\frac1{\ell(Q\sb z)}\int\limits \sb {\tilde I}\left \{ \,\int \limits \sb{S\sb n}
 P\sb {(1-r_n)e^{i\theta}}(w)\,d\mu\sb u (w)\right \}\, \frac{d\theta}
{2\pi}
$$
$$
\ge\frac{|\tilde I|}{2\pi \ell(Q\sb z)}\left(\int \limits \sb{\overline{\Bbb D}}
 P\sb {(1-r_n)e^{i\theta^*}}(w)\,d\mu\sb u (w) -\frac{\eta}4\right)
$$
(for an appropriate $\theta ^*$ with $e^{i\theta^*} \in \tilde I$)
$$
 \ge \frac{1-2^{-\frac{n}2}}{2\pi}(-\log{|u(z\sb 0)|}-\frac{\eta}4
-\frac{\eta}4)
$$
$$
\ge  -\frac1{2\pi}\log{|u(z\sb 0)|}-\eta.
$$
We have shown that
$$
\left |2\pi \frac{\mu\sb u (Q\sb z)}{\ell(Q\sb z)} + \log {|u(z\sb 0)|}
\right| < \eta.
$$
Since $z$ was an arbitrary point with $d(z,z\sb 0) <M$, 
 this concludes the proof of the theorem.

 
\section{On the size of the closed convex hull of  the interpolating Blaschke
products}
In this section we will use  theorem 3.1 to give 
 proofs of the results stated in the introduction.
The line of argument follows \cite {M-S} and \cite{G-N} and the size of 
coefficients is estimated in a manner suggested by \cite{Oy}.
 We make use of a fixed net of dyadic \lq\lq squares \rq\rq.
 By lemma 2.1 we may
assume that a square in the first level of the net containing any zero has 
$\ell(Q) \le \frac{\pi}{128}$.


\begin{thm}
 Let $B$ be a Blaschke product, continuous on               
$\overline {\Bbb D}\setminus E$ where $E\subset \Bbb T$ has Lebesgue 
measure zero. Then $B$ is contained in $9K$.
\end{thm}
\proof  Let $\mathcal M=\{Q\sb \nu \}$ be the set of maximal dyadic squares which contain
no zero of $B$. We have
$$
\sum \limits \sb {Q\sb {\nu}\in \mathcal M}\ell(Q\sb {\nu}) = 2\pi.
$$

Let $N$ be an integer which will be made as large as required
 and let $z\sb \nu$ be the point at the center of the
 top edge of the square $2^{-N}Q\sb \nu$.
Let $B\sb N \in \mathcal I$ denote the Blaschke product with simple zeros
at the points $\{z\sb \nu\}$ and let
$$
B\sb {N,\sigma}(z)= \prod \limits \sb {\nu \ne \sigma} -\frac{\overline{z\sb \nu}}
{|z\sb \nu|}\left(\frac{z-z\sb \nu}{1-\overline{z\sb \nu}z}\right).
$$
Let $\omega \sb n \equiv 2^{-(N-n)}Q\sb{\sigma}$ and let $a\sim b$ denote
the relation $\frac{a}{b} \to 1$ as $N\to \infty$.  
 We have by lemma 2.4
$$
\log{\frac1{|B\sb {N,\sigma}(z\sb {\sigma})|}}\sim \frac12\sum \limits\sb {\nu\ne\sigma}
\frac{(1-|z\sb \sigma|^2)(1-|z\sb \nu|^2)}{|1-\overline{z\sb \sigma}z\sb \nu|^2}
$$
$$
= \sum \limits \sb {n\ge N}\sum \limits \sb {z\sb \nu \in \omega\sb n\setminus    
\omega\sb {n-1}} 
\frac{(1-|z\sb \sigma|^2)(1-|z\sb \nu|^2)}{|1-\overline{z\sb \sigma}z\sb \nu|^2}
$$ 
$$
\le C\sum \limits \sb {n\ge N}\sum \limits \sb {z\sb \nu \in \omega\sb n\setminus    
\omega\sb {n-1}}2^{-2n}\frac{1-|z\sb \nu|^2}{1-|z\sb \sigma|^2}
$$
$$
\le C'\sum \limits \sb {n\ge N}2^{-n}. 
$$
(Note that the sum over
$z\sb \nu \in \omega \sb n \setminus \omega\sb {n-1}$
is vacuous for large $n$.)
Therefore, for a given $\epsilon >0$, we will have 
$$
|B\sb {N,\sigma}(z\sb \sigma)|= (1-|z\sb {\sigma}|^2)|B'\sb N(z\sb
{\sigma})|
> 1-\epsilon \quad \text{for each}\quad \sigma
$$
 if $N= N(\epsilon)$ is sufficiently large.
Altering the previous notation, we write
$$
B\sb N ^w \equiv \frac{B\sb N -w}{1-\overline{w}B\sb N}.
$$
 By lemma 2.3, there are 
positive numbers
$0<r\sb N <1$ and $0<\lambda \sb N<1$ with $r\sb N\to 1$ and
$\lambda\sb N \to 1$ as $N\to \infty$ such that 
$B\sb N ^w \in \mathcal I$ for all $w$ with $|w|<r\sb N $ and such
that
$B\sb N ^w $ has exactly one zero in each disk $D\sb 
{\nu}=\{z:\rho (z,z\sb \nu)
< \lambda \sb N\}$.
By remark 1 following lemma 2.3  we may
 assume each $D\sb {\nu}$ is contained in the square
 $Q\sb \nu$.

If $z\in \Bbb D \setminus \bigcup \limits \sb j Q\sb j$ then the 
dyadic square $Q$ whose top half contains $z$ has
$$
\frac{\mu \sb {BB\sb N}(Q)}{\ell (Q)}\ge\frac{\mu \sb {B\sb N}(Q)}
{\ell (Q)}\ge 2^{-N}.
$$
Otherwise, $z$ is hyperbolically near some square with zero mass.


Let $\delta>0$ satisfy $\frac1{2\pi}\log{\frac1{\delta}}=2^{-(N+1)}$.
Then by theorem 3.1 we have
$$
(*) \qquad\frac{BB\sb N +a}{1+\overline{a}BB\sb N} \in
 \mathcal F \qquad\text{for each}\quad a
 \text{ with } |a|= \delta. 
$$
Now choose $w\in \Bbb D$ with $r\sb N ^2<|w|<r\sb N$. There is $N'$ depending
only on $\lambda \sb N$ such that
$$
\frac{\mu \sb {BB\sb N^w}(Q)}{\ell (Q)}\ge 2^{-N'}
$$ 
for all dyadic squares $Q$ with $$T(Q)\subset \Bbb D\setminus\bigcup \limits
\sb j Q\sb j.$$ 
And since all zeros of $B\sb N ^w$ are contained in the  disks
$\{z:\rho(z,z\sb \nu)<\lambda \sb N\}$ it is clear that any point in 
$\bigcup \limits \sb j Q\sb j $ is  near some square with zero mass. 

So with $\frac1{2\pi}\log{\frac1{\delta'}}=2^{-(N'+1)}$ we have
$$
(**)\qquad\frac{BB\sb N^w +c}{1+\overline{c}BB\sb N^w} \in \mathcal F, 
\qquad\text{for each}\quad c \text{ with } |c|= \delta' . 
$$
Fix $a$ with $|a|=\delta$, and let $\frac{BB\sb N +a}{1+\overline{a}BB\sb N}
=u \in \mathcal F$. Then we have 
$$
BB\sb N = -a+(1+|a|)(1-|a|)(u+\overline{a}u^2 +\overline{a}^2u^3+\dots)\,.
$$
By lemma 2.1 we know that $\cal F$ is contained in the uniform
 closure of $\cal I$, so that
$$
BB\sb N \in (1+2|a|)K\subset3K.        
$$   
Similarly, $(**)$ implies
$$
BB\sb N ^w \in (1+2|c|)K\subset 3K.
$$
Now we  also have
$$
BB\sb N ^w= -wB +(1+|w|)[(1-|w|)(BB\sb N +\overline{w}B(B\sb N) ^2
+\overline{w}^2B(B\sb N)^3+\dots)].
$$
By lemma 2.1 again, we know that each term $B(B\sb N)^n$ is contained
in $3K$,
so $-wB\in 9K$ and since $|w|\to 1$ as $N\to \infty$, we have $B\in 9K$.
\\
\\
Now we factor a general Blaschke product into two factors in such a
way
that a version of the above argument may be applied to each.

\begin{thm}  If $B$ is a Blaschke product defined in $\Bbb D$, then
$B\in 27K$. 
\end{thm}
\proof
Let B be a Blaschke product with zero set $\{w\sb n\}$. We may assume
 that $\{w\sb n\}\cap \partial T(Q)=\emptyset$ for every $Q$ in a 
fixed dyadic net. It is clear that any Blaschke product is the uniform
limit
of products with this property.
By lemma 2.1 we may assume that all the zeros satisfy
$|w\sb n| >1-\frac{\pi}{128}$ and  further, that each dyadic square
$Q$ with $\ell(Q)= \frac{\pi}{128}$ satisfies
$$
\frac{\mu \sb B(Q)}{\ell(Q)} < \epsilon \sb 0.
$$
 
Let $\{Q\sb j ^1\}= \mathcal G_1$ be the set of maximal dyadic squares with
$$
\frac{\mu \sb B (Q\sb j ^1)}{\ell (Q \sb j ^1)}>2^{M}\epsilon\sb 0,
$$
where $M>0$ will later be chosen as large as needed and $\epsilon \sb 0>0$ as
 small as needed.   We have 
$$
\sum \limits \sb{\mathcal G \sb 1}\ell (Q\sb j^1)\le 2\pi2^{-M}.
$$
Now let $\{V\sb k ^1\}$ be the set of dyadic squares which are
maximal among  dyadic squares contained in  some $Q\sb j ^1 \in \mathcal G
\sb 1$
with respect to the property that
$$\frac{\mu(V\sb k ^1)}{\ell (V\sb k ^1)}<\epsilon \sb 0\,.$$

Because $|B| \to  1$ non-tangentially almost everywhere, we have  
$$
\sum \limits\sb{V\sb k ^1 \subset Q\sb j ^1}\ell (V\sb k ^1) =
 \ell (Q\sb j ^1)
\qquad \text{for each}\quad Q\sb j ^1 \in \cal G \sb 1.
$$
To see this, let $Q$ be a square such that $z\in \Bbb D$ is contained in
$T(Q)$. We then have
$$
\log {|B(z)|} \le -C\sum \limits \sb {w\sb n \in Q}
\frac{(1-|z|^2)(1-|w\sb n|^2)}{|1-\overline{w\sb n}z|^2}
$$
$$
\le -C'\sum \limits \sb {w\sb n \in Q}\frac{1-|w\sb n|}{1-|z|}
$$
$$
\le -C'\frac{\mu \sb B(Q)}{\ell(Q)}
$$
and this implies what was claimed.

Let
$$
R\sb j ^1 = Q\sb j ^1 \setminus \bigcup \limits \sb{V\sb k ^1
 \subset Q\sb j ^1}
V\sb k ^1 \qquad \text{for each}\quad Q\sb j ^1 \in \cal G\sb 1.
$$



Define $\cal G\sb 2 =\{Q\sb j ^2\}$ to be the
 set of dyadic squares which are
  maximal among squares contained in  $V\sb k^1 $ for some $k$
 with respect to the property that
$$\frac{\mu(Q\sb j ^2)}{\ell (Q\sb j ^2)}\ge 2^{M}\epsilon \sb 0\,.$$ 
We have 
$$
\sum \limits \sb{Q\sb j ^2 \subset V\sb k ^1}\ell (Q\sb j ^2)\le
 2^{-M}\ell (V \sb k ^1),\qquad \text{for each}\quad k.
$$
We form the set of  squares $\{V\sb k ^2\}$ and regions $R\sb j ^2$ as
before
and continue to obtain $Q\sb j ^n$, $ R\sb j ^n$ and $V\sb k ^n$.
 If $2^{M}\epsilon\sb  0
<\frac12$ our assumptions imply that all zeros of $B$ are contained in the
 interiors of the regions $R\sb j ^n$.
Let $\sigma$ and $\nu$ index the pairs $(n,k)$. 
 Choose an integer $N>0$ and let
 $B\sb N \in \cal I$ be the product with one zero  
$z\sb \nu= z\sb {(n,k)}$ 
at the center of the top edge of each square $2^{-N}V\sb k^n$ for all
 $n$ and $k$. 
 We will also write $V\sb k^n= V\sb {\nu}$. Let
$$
B\sb {N,\sigma}\equiv \prod \limits \sb {\nu\ne \sigma}-\frac{\overline{z\sb \nu}}
{|z\sb \nu|}\frac{z-z\sb \nu}{1-\overline{z\sb \nu}z} 
$$

We claim that given $\epsilon>0$, if $N$  and $M$ are sufficiently
large, then 
$$
|B\sb {N,\sigma}(z\sb \sigma)|\ge1-\epsilon.
$$
Let  $S\sb n$ denote the square $2^{-(N-n)}V\sb {\sigma}$.
Then we have
$$
\log{\frac1{|B\sb {N,\sigma}(z\sb{\sigma})|}}\sim \sum \limits\Sb z\sb \nu \in V\sb \sigma\\
 \nu\ne\sigma \endSb
\frac{(1-|z\sb \sigma|^2)(1-|z\sb \nu|^2)}{|1-\overline{z\sb \sigma}z\sb \nu|^2}
+
\sum \limits \sb {n> N}\sum \limits \sb {z\sb \nu \in S\sb n\setminus
S\sb {n-1}}
\frac{(1-|z\sb \sigma|^2)(1-|z\sb \nu|^2)}{|1-\overline{z\sb \sigma}z\sb \nu|^2}.
$$
As in the 
previous proof the sum on the right is as small as we like if $N$
is large enough. The sum on the left is 
$$
\le C\sum \limits \sb {n\le N}\sum \limits \sb {z\sb \nu \in S\sb n\setminus
S\sb {n-1}}2^{-2n}\left(\frac{1-|z\sb \nu|^2}{1-|z\sb \sigma|^2}\right)
$$

$$
\le C' \frac{2^N}{\ell(V\sb \sigma)}\sum \limits \sb {n\le N}\sum \limits \sb
{z\sb \nu \in S\sb n\setminus
S\sb {n-1}}2^{-2n}(1-|z\sb \nu|^2)  
$$

$$
\le C'' \frac{2^N}{\ell(V\sb \sigma)}\sum \limits \sb {n\le N}2^{-2n}
\cdot2^{-(M+N)}\ell(V\sb \sigma)
$$

$$
\le C' \,2^{-M}\sum \limits \sb {n\le N}2^{-n}
$$
and therefore is as small as we like if $M$ is sufficiently large.

As in the proof of the previous theorem  there are positive numbers
$0<r\sb N <1$ and $0<\lambda \sb N<1$ with $r\sb N\to 1$ and
$\lambda\sb N \to 1$ as $N\to \infty$ such that
$$
B\sb N ^w \equiv \frac{B\sb N -w}{1-\overline{w}B\sb N}\in \cal I 
\qquad \text{for each}\quad w \text{ with } |w|<r\sb N
$$
and $B\sb N ^w$ has exactly one zero in each disk $\{z:\rho (z,z\sb \nu)
< \lambda \sb N\}$. By the remark following lemma 2.3 we may assume
that each such disk is contained in the
 square $Q\sb \nu$. 
Taking $M$ to be sufficiently larger than $N$ we may also assume that each 
$D\sb \nu$ is hyperbolically far from any region $R\sb j ^n$. 

For a fixed $Q\sb j ^ n$ let $\Omega \sb j ^ n$ be the unique dyadic square
 containing $Q\sb j ^n $ with $\ell (\Omega \sb j ^ n )= 2\ell (Q\sb j ^n)$.
Notice that by the maximality of the $Q\sb j ^n$, if there are two adjacent
$Q\sb j ^n$ of the same size, say $Q\sb k ^n$ and $Q\sb l ^n$ then
$\Omega \sb k ^n \ne \Omega \sb l ^n$.
                
Let 
$$
\cal G' \sb n = \{\Omega \sb j ^n :Q\sb j ^n \in \cal G \sb n\}
$$
and write
$$
\cal G' \sb n=\bigcup \limits \sb {i=1}^{\infty} \cal H\sb i
$$
where $\cal H\sb 1$ is the set of maximal dyadic squares in $\cal G' \sb n$
and, for $n>1$ , $ \cal H\sb n$ is the set of maximal squares in 
$$
\cal G' \sb n \setminus \bigcup \limits \sb {i=1}^{n-1} \cal H\sb i.
$$
In the terminology of \cite{G} we have divided each set of squares
$\mathcal G'\sb n$ into generations $\mathcal H\sb i$.
Note that if $\Omega \sb j ^n \in \mathcal H \sb i$ then
$$
\sum \limits \sb {\{\Omega \sb k ^n \in \mathcal H\sb {i+2}:\Omega \sb k ^n
\subset \Omega \sb j ^n\}} \ell (\Omega \sb k ^n )
\le
\frac14 \ell (\Omega \sb j ^n ).
$$

Define
$$
\cal G\sb {n,0}=\bigcup \limits \sb {i=1}^{\infty}\{Q\sb j ^n \in \cal G\sb n:
\Omega \sb j ^n \in \cal H\sb {2i-1}\}
$$
and
$$
\cal G\sb {n,1}=\bigcup \limits \sb {i=1}^{\infty}\{Q\sb j ^n \in \cal G\sb n:
\Omega \sb j ^n \in \cal H\sb {2i}\},
$$
and define
$$
W\sb {n,\alpha}=\bigcup \limits \sb {Q\sb j ^n \in \cal G\sb {n,\alpha}}
R\sb j ^n \qquad \alpha=0,1
$$
and
$$
U\sb \alpha = \bigcup \limits \sb {n=1}^{\infty}W\sb {n,\alpha}\qquad \alpha=
0,1.
$$
 We will now use these definitions to
factor
the product $B$. 

Let
$$
B\sb \alpha = B|\sb {U\sb \alpha}, \quad \alpha=0,1
$$
and let 
$$
B\sb {N,n,\alpha}= \prod \limits \sb {\{\nu=(n,k),j:V\sb k ^n \subset
G\sb j ^n \in \mathcal G\sb {n,\alpha}\}}
 -\frac{\overline{z\sb \nu}}
{|z\sb \nu|}\left(\frac{z-z\sb \nu}{1-\overline{z\sb \nu}z}\right), 
\qquad \alpha=0,1
$$  
and
$$
B\sb {N,\alpha} = \prod \limits \sb n B\sb {N,n,\alpha}\,\,,\qquad \alpha=0,1.
$$
Then $B= B\sb 0 B\sb 1 $  and $B\sb N = B\sb {N,0}B\sb {N,1}$.

Let
$$
B\sb {n,\alpha}^* =
B\sb {\alpha}|\sb {W\sb {n,\alpha}}\cdot B\sb {N,n,\alpha}
$$
and let
 $$
B\sb {\alpha} ^*= \prod \limits \sb n B\sb {n,\alpha}^*
    = B\sb {\alpha}B\sb {N,\alpha}.
$$
 We claim that 
$$
\frac{B\sb \alpha ^*-\delta}{1-\overline{\delta}B\sb \alpha ^*}\in \cal F, 
\qquad \alpha=0,1
$$
for all $\delta$ with $\frac1{2\pi}\log{\frac1{|\delta|}}
=\frac78  2^{-N}$. 
This will follow from theorem 3.1  once we
 show that every point
in $\Bbb D$ is within bounded hyperbolic distance of some dyadic square
$Q$  with
$$
\frac{\mu\sb {B\sb \alpha ^*}(Q)}{\ell(Q)}\notin[\frac342^{-N},2^{-N}).  
$$

Let $\alpha$ be fixed.
If $Q\sb j ^n \in \cal G\sb {n,\alpha}$ then
$$
\frac{\mu \sb {B\sb {n, \alpha} ^*}(\Omega \sb j ^n)}
{\ell(\Omega \sb j ^n)} \le \left(\frac{2^{-N}+2^M\epsilon\sb 0}2\right) 
+\left(\frac14\right)\left(\frac{2^{-N}+2^M\epsilon\sb 0}2\right)
 +\left(\frac14\right)^2\left(\frac{2^{-N}+2^M\epsilon\sb 0}2\right)+
\cdots
$$
$$
\le \frac46(2^{-N}+2^M\epsilon\sb0).
$$
Here the factor $\frac14$ is provided by the \lq \lq odd-even\rq\rq\
splitting within each set $\mathcal G \sb n$.

Now summing over all the sets $\mathcal G\sb n$ we have
$$
\frac{\mu \sb {B\sb \alpha ^*}(\Omega \sb j ^n)}{\ell (\Omega \sb j ^n)}
\le (1+ 2^{-M}+2^{-2M}+\cdots)\cdot
\frac46(2^{-N}+2^M\epsilon\sb 0)<\frac342^{-N}
$$
if $\epsilon \sb 0$ is small.

Therefore if $z$ is not in any $R\sb j ^n$ with $Q\sb j ^N\in
 \cal G\sb {n,\alpha}$ 
then it is hyperbolically near the top half of a dyadic square $Q$ with
$\frac {\mu \sb {B\sb {\alpha} ^*}(Q)}{\ell(Q)}<\frac342^{-N}$.
This is because the point $z$ is inside some square $V\sb {\nu}$ but
outside
any $R\sb j ^n$. Moving a fixed hyperbolic distance away from the zero
$z\sb {\nu}$ if necessary, there are only squares with
$\frac{\mu}{\ell}< \frac34 2 ^{-N}$ directly below.


 On the other hand,
if $z\in R\sb j ^n$ with $Q\sb j ^n \in \cal G\sb {n,\alpha}$  
 then $z$ is contained in the top
half of some dyadic square $Q$ with $\frac {\mu \sb {B\sb {\alpha} ^*}(Q)}
{\ell(Q)}\ge 2^{-N} $. The claim is proved.

Following the  proof of the previous theorem, we may replace $B\sb N$
by $B\sb N ^w=\frac{B\sb N -w}{1-\overline{w}B\sb N}$ where
$r\sb N ^2<|w|<r\sb N$. The zeros of $B\sb N $ and $B\sb N^w$ are in 
a natural one to one correspondence since each product has exactly one zero
in the disk
$$
\{z:\rho(z,z\sb \nu)<\lambda\sb N\}.
$$
 We let $B\sb {N,\alpha}
^w$ be the factors of $B\sb N^w$ corresponding to the factors 
$B\sb {N,\alpha}$ of $B\sb N$. Then there exists an $N'$ depending only on
$\lambda \sb N$ such that
$$
\frac{B\sb {\alpha}B\sb {N,\alpha}^w - \eta}{1-\overline{\eta}B\sb
 {\alpha}B\sb {N,\alpha}^w}
\in \cal F, \qquad \alpha=0,1
$$
for all $\eta$ with $\frac1{2\pi} \log{\frac1{|\eta |}}=\frac78 2^{-N'}$.
This follows by a repetition of the previous calculation.

We obtain as before
$$
B\sb {\alpha}B\sb {N,\alpha} \in (1+2|\delta|)K \quad \text{ and } \quad
B\sb {\alpha}B\sb {N,\alpha}^w \in (1+2|\eta|)K
$$
therefore
$$ 
BB\sb N \in (1+2|\delta|)^2K \quad \text{ and } \quad
BB\sb N^w \in (1+2|\eta|)^2K.
$$
It follows that
$$
-wB=BB\sb N^w-(1-|w|^2)(BB\sb N +\overline{w}B(B\sb N)^2 +
\overline{w}^2B(B\sb N)^3 + \dots) 
$$
and therefore, by lemma 2.1, that $-wB \in 27K$. Since $|w|\to1$ as $N\to \infty$ we have
$B\in 27K$. 
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