\section{Pocket calculus}\label{sec:pockets}

Throughout this section $(W,B,w_0)$ is a triple as in Section~\ref{sec:criterion}, with $k=\deg B\ge1$, zeros $a_1,\dots,a_k$ and
$z_0=W(w_0)$; $v$, $\sigma$, $\gamma$ and $N$ are as in Section~\ref{sec:criterion}. For $a\in\mathbb D$ and $|w|=1$ we write
$P_a(w)=(1-|a|^2)/|w-a|^2$, so that $\sigma=\sum_jP_{a_j}$ on the circle, and $\omega$ for the measure
$P_{w_0}(e^{i\theta})\,d\theta/2\pi$, the harmonic measure of $z_0$ in $\Omega=W(\mathbb D)$ pulled back to the circle. The
triple is \emph{feasible}\label{term:feasible} if $((W-z_0)B)'$ has no zero on $\{|w|\le1\}$; by Proposition~\ref{prop:criterion} this holds if and
only if $\gamma\neq0$ on the circle and $\operatorname{wind}(\gamma)=-k$. For a feasible triple, $B(w_0)\neq0$, since
$G'(w_0)=W'(w_0)B(w_0)\neq0$.

\subsection{Pockets}\label{sec:pocketdef}
On the circle, $w=e^{i\theta}$, the point $\zeta=W(w)$ runs along $\partial\Omega$ in the positive sense with tangent
$iwW'(w)$, so the outward unit normal is $n=wW'/|W'|$ and
\[
  v=\frac{\zeta-z_0}{|W'|\,n},\qquad \frac{d}{d\theta}|\zeta-z_0|^2=2|W'|^2\operatorname{Im}v .
\]
Thus $\alpha:=|\arg v|\in[0,\pi]$ is the angle between $\zeta-z_0$ and the outward normal at $\zeta$, and $\operatorname{Re}v<0$
if and only if $\alpha>\pi/2$; we then call $\zeta$ \emph{back-facing}\label{term:backfacing}.

\begin{definition}\label{def:pocket}
A \emph{pocket} is a point $w$ of the circle at which $v(w)$ is a negative real number; its \emph{depth} is $x(w):=-v(w)>0$, and
$Z$ denotes the set of pockets. The \emph{sign} $\varepsilon(w)\in\{+1,0,-1\}$ of a pocket is the sign change of
$\operatorname{Im}v$ at $w$ as $\theta$ increases: $+1$ if $\operatorname{Im}v$ passes from positive to negative (then
$|\zeta-z_0|$ has a local maximum along $\partial\Omega$ at $\zeta=W(w)$), $-1$ if it passes from negative to positive (a local
minimum), and $0$ if it does not change sign. A pocket is \emph{on} for $\sigma$ if $\sigma x>1$ and \emph{off} if $\sigma x<1$.
\end{definition}

\begin{lemma}[frames]\label{lem:frames}
Let $\varphi_{w_0}(u)=(u+w_0)/(1+\bar w_0u)$, and let $\widetilde W$, $\widetilde B$ be as in the proof of Corollary~\ref{cor:BF}, with
quantities $\widetilde v$, $\widetilde\sigma$ for the triple $(\widetilde W,\widetilde B,0)$. On the circle,
\[
  \widetilde v(u)=P_{w_0}(\varphi_{w_0}(u))\,v(\varphi_{w_0}(u)),\qquad \widetilde\sigma(u)=\sigma(\varphi_{w_0}(u))/P_{w_0}(\varphi_{w_0}(u)),
\]
and the same holds for each $P_{a_j}$. Hence the products $\sigma x$ and $P_{a_j}x$, the set of pockets and their signs do not
depend on the \emph{frame}\label{term:frame}, that is, on whether we work with $(W,B,w_0)$ or with $(\widetilde W,\widetilde B,0)$. The \emph{base-$0$ depth} $x_0:=P_{w_0}\,x=2\pi|\zeta-z_0|\,d\omega/ds$ ($s$ = arc length on $\partial\Omega$) is the
depth in the frame $(\widetilde W,\widetilde B,0)$.
\end{lemma}

\begin{proof}
Write $\varphi=\varphi_{w_0}$. On the circle $u\varphi'(u)/\varphi(u)=|\varphi'(u)|=1/P_{w_0}(\varphi(u))$, since $\varphi$ is an orientation-preserving automorphism of the
disk. Then $\widetilde v(u)=(W(\varphi(u))-z_0)/(uW'(\varphi(u))\varphi'(u))=v(\varphi(u))\,\varphi(u)/(u\varphi'(u))$, and
$\widetilde\sigma(u)=u\varphi'(u)B'(\varphi(u))/B(\varphi(u))=\sigma(\varphi(u))\,u\varphi'(u)/\varphi(u)$; the factor is positive, so the pockets and
the signs are preserved (the parametrisation keeps its orientation). The same computation applies to each factor $B_{a_j}$,
since $B_{a_j}\circ\varphi$ is a unimodular multiple of $B_{\varphi^{-1}(a_j)}$. Finally $d\omega/ds=P_{w_0}/(2\pi|W'|)$ and
$x=|\zeta-z_0|/|W'|$ at a pocket.
\end{proof}

\begin{lemma}\label{lem:Zfinite}
If $\Omega$ is a disk centred at $z_0$, there are no pockets. Otherwise $\operatorname{Im}v$ has finitely many zeros on the circle;
in particular $Z$ is finite and every sign is defined.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:frames} we may take $w_0=0$, $W\in\mathcal S$. Then $v=W/(wW')$ is holomorphic and zero-free near the
closed disk, with $v(0)=1$. If $\operatorname{Im}v=0$ on the whole circle, the harmonic function $\operatorname{Im}v$ vanishes in
$\mathbb D$, so $v\equiv1$, $wW'=W$ and $W(w)=w$: $\Omega$ is the unit disk and $v=1$ has no pocket. Otherwise the
real-analytic function $\operatorname{Im}v$ on the circle is not identically zero and has finitely many zeros.
\end{proof}

\subsection{The pocket formula}\label{sec:pocketformula}

\begin{theorem}[pocket formula]\label{thm:pocket}
Let $\hat\sigma>0$ be any continuous function on the circle with $\hat\sigma x\neq1$ on $Z$. Then $1+\hat\sigma v$ has
no zero on the circle and
\[
  \operatorname{wind}(1+\hat\sigma v)=\sum_{w\in Z,\ \hat\sigma(w)x(w)>1}\varepsilon(w).
\]
\end{theorem}

\begin{proof}
Put $\hat\gamma=1+\hat\sigma v$. Then $\hat\gamma=0$ means $v=-1/\hat\sigma$, a pocket with $\hat\sigma x=1$, which is excluded. Since $\hat\sigma>0$, $\hat\gamma$ is real exactly where $v$ is
real: where $v>0$ we have $\hat\gamma>1$, and at a pocket $\hat\gamma=1-\hat\sigma x$, which is negative if and only if $\hat\sigma x>1$. So
$\hat\gamma$ meets the negative real axis only at the finitely many pockets with $\hat\sigma x>1$, and its winding number about $0$ is the sum, over these
points, of the local crossing numbers: $+1$ if $\operatorname{Im}\hat\gamma$ passes from positive to negative, $-1$ for the opposite
passage, and $0$ if it does not change sign. As $\operatorname{Im}\hat\gamma=\hat\sigma\operatorname{Im}v$, this local number is
$\varepsilon$.
\end{proof}

With Proposition~\ref{prop:criterion}: a triple is feasible if and only if no pocket has $\sigma x=1$ and the signs of the on
pockets add up to $-k$. So, for $\sigma$ with $\sigma x\neq1$ on $Z$, feasibility depends on $\sigma$ only through the set of pockets that $\sigma$ switches on.

\begin{corollary}\label{cor:constant}
Let $w_0=0$. For every $c>0$ such that $1/c$ is not a depth, $\operatorname{wind}(1+cv)\ge0$. Consequently, in any frame, for
every $T>0$ that is not a base-$0$ depth,
\[
  \sum_{w\in Z,\ x_0(w)>T}\varepsilon(w)\ \ge\ 0 .
\]
In particular the signs of the pockets of largest base-$0$ depth add up to a non-negative number, and a unique deepest pocket has
$\varepsilon\ge0$ (a local maximum of $|\zeta-z_0|$, or a point where $\operatorname{Im}v$ does not change sign).
\end{corollary}

\begin{proof}
$1+cv=(wW'+c(W-z_0))/(wW')$. The numerator is holomorphic near the closed disk, has no zero on the circle and vanishes at
$w=0$; the denominator has exactly one zero in the closed disk, at $0$. So $\operatorname{wind}(1+cv)$ is the number of zeros of
the numerator in $\mathbb D$ minus one, which is $\ge0$. Apply Theorem~\ref{thm:pocket} with $\hat\sigma=c=1/T$ in the frame
$(\widetilde W,\widetilde B,0)$ of Lemma~\ref{lem:frames}; for the last claim take $T$ just below the largest base-$0$ depth.
\end{proof}

In a frame with $w_0\neq0$ the numerator $wW'+c(W-z_0)$ takes the value $w_0W'(w_0)\neq0$ at $w_0$, while $wW'$ still
vanishes at $0$, and the argument gives only $\operatorname{wind}\ge-1$; this is why Corollary~\ref{cor:constant} uses base-$0$
depths.

\begin{proposition}[guard]\label{prop:guard}
Let the triple be feasible and let $\mathcal O$ be the set of its on pockets. Then some pocket outside $\mathcal O$ has $\varepsilon=+1$ and base-$0$
depth at least $\min_{\mathcal O}x_0$.
\end{proposition}

\begin{proof}
By Theorem~\ref{thm:pocket}, $\sum_{\mathcal O}\varepsilon=-k\le-1$, so $\mathcal O\neq\emptyset$. Let $x_s=\min_{\mathcal O}x_0$ and choose $T<x_s$ such that no
base-$0$ depth lies in $[T,x_s)$. Every pocket of $\mathcal O$ has $x_0>T$, so by Corollary~\ref{cor:constant}
\[
  \sum_{x_0>T,\ w\notin {\mathcal O}}\varepsilon=\sum_{x_0>T}\varepsilon-\sum_{\mathcal O}\varepsilon\ \ge\ 0+k\ \ge1,
\]
and a pocket outside $\mathcal O$ with $x_0>T$, hence $x_0\ge x_s$, has $\varepsilon=+1$.
\end{proof}

Proposition~\ref{prop:guard} gives one off pocket of sign $+1$ at least as deep as the shallowest on pocket; it does not give
one for each on pocket of sign $-1$.

\subsection{Independence}\label{sec:independence}

\begin{lemma}[pocket independence]\label{lem:independence}
Let $Z=Z_0\cup Z_1\cup\dots\cup Z_k$ be a partition, and write $P_j=P_{a_j}$. Assume
\begin{enumerate}
\item[(I1)] for each $j\ge1$: $\sum\{\varepsilon(w):w\in Z_j,\ P_j(w)x(w)>1\}=-1$;
\item[(I2)] for each $j\ge1$ and $w\in Z_j$: $P_jx\neq1$, and $P_jx<1$ implies $\sigma x<1$; for $w\in Z_0$: $\sigma x<1$.
\end{enumerate}
Then the triple is feasible, with gain $1/|B(w_0)|=\prod_j1/|B_{a_j}(w_0)|$.
\end{lemma}

\begin{proof}
Since $\sigma\ge P_j$, a pocket of $Z_j$ with $P_jx>1$ is on; by (I2) all other pockets are off. So $\sigma x\neq1$ on $Z$, the on
pockets are those of the sets $\{w\in Z_j:P_jx>1\}$, and by (I1) and Theorem~\ref{thm:pocket} $\operatorname{wind}(\gamma)=-k$.
Proposition~\ref{prop:criterion} gives feasibility.
\end{proof}

By Lemma~\ref{lem:frames} the hypotheses do not depend on the frame. (I2) holds if, for each $j\ge1$,
$P_jx\neq1$ on $Z_j$ and $\sum_{l\neq j}P_{a_l}(w)<1/x(w)-P_j(w)$ at every $w\in Z_j$ with $P_j(w)x(w)<1$, and if $\sigma<1/x$ on $Z_0$;
here $P_{a_l}(w)\le2(1-|a_l|)/|w-a_l|^2$. Lemma~\ref{lem:independence} combines zeros, and their gains multiply, provided that for each $j$
adding the kernels $P_{a_l}$, $l\neq j$, to $P_j$ switches on no pocket of $Z_j$ that is off for $P_j$. Section~\ref{sec:ownership} proves a converse direction: in
every Blaschke model each zero switches on a pocket of its own.

\subsection{The gain identity and the sine bound}\label{sec:gain}

\begin{proposition}[gain identity]\label{prop:gain}
\textup{(a)} If the triple is feasible, then
\[
  \log\frac1{|B(w_0)|}=-\int\log|\gamma|\,d\omega .
\]
\textup{(b)} For every $B$ as in \textup{(M2)} with $B(w_0)\neq0$, feasible or not,
\[
  -\int\log|\gamma|\,d\omega=\log\frac1{|B(w_0)|}-\sum_bg_{\mathbb D}(w_0,b),
\]
where $b$ runs over the zeros of $N$ in $\mathbb D$ with multiplicity and $g_{\mathbb D}$ is the Green function of the disk. So
$-\int\log|1+\sigma v|\,d\omega\le\log(1/|B(w_0)|)$, with equality if and only if $N$ has no zero in $\mathbb D$.
\end{proposition}

\begin{proof}
$N=B+((W-z_0)/W')B'$ is holomorphic near the closed disk, $N(w_0)=B(w_0)$, and $|N|=|B||\gamma|=|\gamma|$ on the circle.
(b) is Jensen's formula for $N$ at $w_0$, written with the harmonic measure $\omega$ of $w_0$ (zeros of $N$ on the circle make
$\log|\gamma|$ integrable and contribute nothing). (a) is the case without zeros in the closed disk.
\end{proof}

\begin{theorem}[sine bound]\label{thm:sin}
Let
\[
  \mathcal I(\Omega,z_0):=\int_{\{\alpha>\pi/2\}}\log\frac1{\sin\alpha}\,d\omega,
\]
the harmonic measure of the back-facing part of $\partial\Omega$, seen from $z_0$, weighted by $\log(1/\sin\alpha)$. Then
$\mathcal I(\Omega,z_0)<\infty$, and every feasible triple on $(W,w_0)$, of any degree, satisfies $\log(1/|B(w_0)|)\le {\mathcal I}(\Omega,z_0)$.
\end{theorem}

\begin{proof}
Since $\sigma>0$: where $\operatorname{Re}v\ge0$ we have $|1+\sigma v|\ge\operatorname{Re}(1+\sigma v)\ge1$, and where
$\operatorname{Re}v<0$, $|1+\sigma v|$ is at least the distance from $0$ to the line $1+\mathbb Rv$, which is
$|\operatorname{Im}v|/|v|=\sin\alpha$. Insert this into Proposition~\ref{prop:gain}(a). Finiteness: $\log(1/\sin\alpha)=
\log|v|-\log|\operatorname{Im}v|$, $|v|$ is bounded above and below on the circle, and $\log|\operatorname{Im}v|$ is integrable
because $\operatorname{Im}v$ is real-analytic with finitely many zeros (Lemma~\ref{lem:Zfinite}; if $\Omega$ is a disk about
$z_0$ the back-facing set is empty).
\end{proof}

\begin{remark}\label{rem:sin}
(1) If $\Omega$ is star-shaped with respect to $z_0$, then $\operatorname{Re}v\ge0$ everywhere, $\mathcal I(\Omega,z_0)=0$, and no feasible
triple on $(W,w_0)$ has $1/|B(w_0)|>1$; in particular none has $k\ge1$. (2) If $\alpha<\pi$ on the whole of $\partial\Omega$, there is no pocket; then
$\operatorname{wind}(\gamma)=0$ for every $\sigma$, and no feasible triple with $k\ge1$ exists on $(W,w_0)$, however large
$\mathcal I(\Omega,z_0)$ is. So $e^{\mathcal I}$ can be far from the largest gain on $(W,w_0)$.
\end{remark}