\subsection{Counting the zeros of \texorpdfstring{$(WB)'$}{(WB)'} from the circle}\label{sec:count}

Theorem~\ref{thm:signrule} has a quantitative form: an exact count of the zeros of $(WB)'$ in $\mathbb D$, read off the boundary, for
every finite Blaschke product $B$. Let $W(w)\not\equiv w$, $P$, $u$, $\vartheta$, $g$ and $Z_-$ be as in Section~\ref{sec:ratext}. Let $B$ be a
finite Blaschke product of degree $k\ge1$, with zeros anywhere in $\mathbb D$ (repetitions allowed). Put $F=WB$, $\sigma=wB'/B$ and
$\chi=g+\sigma=wF'/F$, and let $N_{\mathbb D}(F')$ be the number of zeros of $F'$ in $\mathbb D$, counted with multiplicity. For a Blaschke model with
base point $0$, the pockets of Section~\ref{sec:pockets} are the points of $Z_-$, and a pocket is on if and only if $\chi>0$ there
(see the text after Theorem~\ref{thm:count}). On
$\mathbb T\setminus P$ we have $\operatorname{Im}\chi=-u'$, as $\sigma$ is real there. For $\zeta\in\mathbb T\setminus P$ write
\[
  \chi(\zeta e^{it})=c_\zeta t^{m_\zeta}+O(t^{m_\zeta+1})\qquad(t\in\mathbb R,\ t\to0,\ c_\zeta\neq0).
\]
As $F(\zeta)\neq0,\infty$, $m_\zeta\ge0$ is the order of the zero of $F'$ at $\zeta$. If $u'$ vanishes at $\zeta$, let $k_\zeta\ge1$ be
the order of this zero; it is finite, since $u$ is constant on no arc of $\mathbb T\setminus P$. Let $\varepsilon_\zeta=+1$, $-1$ or $0$
according as $u$ has a strict local maximum, a strict local minimum, or neither at $\zeta$. In the last case $u'$ has one sign $s_\zeta$
on both sides of $\zeta$. So $k_\zeta$ is odd if and only if $\varepsilon_\zeta\neq0$. Since $\operatorname{Im}\chi=-u'$, we have
$k_\zeta\ge m_\zeta$, and $c_\zeta$ is real if $k_\zeta>m_\zeta$. The set
\[
  Z_\chi=\{\zeta\in\mathbb T\setminus P:\ \chi(\zeta)\in(-\infty,0]\}
\]
lies in $Z_-$, since there $u'=0$ and $\vartheta'=\chi-\sigma<0$. It contains the zeros of $F'$ on $\mathbb T\setminus P$, and no zero
of $W'$ (there $\chi=\sigma>0$). The index defined next is the contribution of $\zeta$ to minus the winding number of
$\chi$ along the circle indented into the disk at the points of $Z_\chi$ and $P$; the indices add up to $k-N_{\mathbb D}(F')$
(Theorem~\ref{thm:count}(i)). For $\zeta\in Z_\chi$ define the \emph{index}\label{term:index}
\[
  \iota_\zeta=\begin{cases}
    m_\zeta/2+\varepsilon_\zeta & \text{if $m_\zeta$ is even and $c_\zeta<0$},\\
    m_\zeta/2 & \text{if $m_\zeta$ is even and $c_\zeta\notin(-\infty,0)$},\\
    (m_\zeta+\varepsilon_\zeta)/2 & \text{if $m_\zeta$ is odd and $\varepsilon_\zeta\neq0$},\\
    (m_\zeta+s_\zeta\operatorname{sgn}c_\zeta)/2 & \text{if $m_\zeta$ is odd and $\varepsilon_\zeta=0$ (then $c_\zeta$ is real)}.
  \end{cases}
\]
If $m_\zeta=0$, that is $\chi(\zeta)<0$, then $\iota_\zeta=\varepsilon_\zeta$: $-1$ at a minimum of $u$ and $+1$ at a maximum. Finally put
\[
  \operatorname{wt}(W)=\sum_{\zeta\in Z_-}\frac{k_\zeta+\varepsilon_\zeta}2 .
\]
Every term is a non-negative integer, and $\operatorname{wt}(W)$ depends on $W$ only.\label{term:wt}

\begin{theorem}\label{thm:count}
Let $W$ and $B$ be as above.
\begin{enumerate}
\item[(i)] $N_{\mathbb D}(F')=k-\sum_{\zeta\in Z_\chi}\iota_\zeta$. In particular $F'\neq0$ in $\mathbb D$ if and only if the indices
  add up to $k$.
\item[(ii)] $\iota_\zeta\le(k_\zeta+\varepsilon_\zeta)/2$ for every $\zeta\in Z_\chi$. Hence every admissible pair $(W,B)$ satisfies
  $\deg B\le\operatorname{wt}(W)$.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) Let $\mathbb T_\rho$ be the circle with inward semicircular indents of radius $\rho$ at the points of $P$ and at the zeros of $\chi$ on
$\mathbb T\setminus P$. $F'$ is meromorphic near the closed disk with poles only at $P$, and $|F'|\to\infty$ at $P$, so it has finitely many zeros in
$\mathbb D$.
For small $\rho$ the region inside $\mathbb T_\rho$ contains all of them, all zeros of $B$, and no pole of $W$. Since $wF'$ and $F$ are
holomorphic there,
\[
  \operatorname{wind}(\chi\circ\mathbb T_\rho)=\operatorname{wind}(wF'\circ\mathbb T_\rho)-\operatorname{wind}(F\circ\mathbb T_\rho)=(1+N_{\mathbb D}(F'))-(1+k).
\]
Around each point $x$ of $Z_\chi\cup P$ let $\mathcal A_x$ be the part of $\mathbb T_\rho$ formed by the arc of $\mathbb T$ from $xe^{-i\eta}$ to the
indent at $x$ (if any), the indent, and the arc to $xe^{i\eta}$. On the rest of $\mathbb T_\rho$, $\chi$ avoids $(-\infty,0]$, so the
principal argument $\operatorname{Arg}\chi$ is continuous there. Put
\[
  \delta_x=\frac1{2\pi}\bigl(\operatorname{Arg}\chi(xe^{i\eta})-\operatorname{Arg}\chi(xe^{-i\eta})-\Delta_{\mathcal A_x}\arg\chi\bigr),
\]
an integer. The terms $\operatorname{Arg}$ telescope around $\mathbb T_\rho$, so $\operatorname{wind}(\chi\circ\mathbb T_\rho)=-\sum_x\delta_x$. Each
$\delta_x$ is constant in $\rho$ and $\eta$, and we evaluate it as $\rho,\eta\to0$.

At a pole $p$ of order $m_p$, $\mathcal A_p$ avoids $(-\infty,0]$: on the circle $|\operatorname{Im}\chi|\to\infty$ at $p$, since
$g(pe^{it})=\frac{im_p}2\cot\frac t2+O(1)$, and on the indent see part~(c) of the proof of Theorem~\ref{thm:signrule}. So
$\delta_p=0$.

Let $\zeta\in Z_\chi$, $m=m_\zeta$, $c=c_\zeta$. On the indent $w-\zeta=\rho\zeta e^{i\theta}$, with $\theta$ decreasing from $-\pi/2$
to $-3\pi/2$, we have $\chi=c\rho^me^{im(\theta-\pi/2)}(1+O(\rho))$, so $\arg\chi$ decreases by $m\pi+O(\rho)$. On the two arcs of the
circle a continuous branch of $\arg\chi$ changes by $O(\eta)$. At the ends, $\chi(\zeta e^{\pm i\eta})=c_\pm\eta^m(1+O(\eta))$ with
$c_\pm=c(\pm1)^m$. If $c_\pm\notin(-\infty,0)$, then $\operatorname{Arg}\chi(\zeta e^{\pm i\eta})\to\operatorname{Arg}c_\pm$. If
$c_\pm<0$, then $c$ is real, so $k_\zeta>m$. Then $\operatorname{Im}\chi(\zeta e^{\pm i\eta})=-u'$ is of order $\eta^{k_\zeta}$, with
the sign opposite to the sign $s_\pm$ of $u'$ on that side, and $\operatorname{Arg}\chi(\zeta e^{\pm i\eta})\to-s_\pm\pi$. So
$\delta_\zeta=(m\pi+A_+-A_-)/2\pi$, with $A_\pm$ these limits. At an extremum $\varepsilon_\zeta=s_-=-s_+$; otherwise $s_-=s_+=s_\zeta$.
\begin{itemize}
\item If $m$ is even, then $c_+=c_-=c$. If $c\notin(-\infty,0)$, then $A_+=A_-$ and $\delta_\zeta=m/2$. If $c<0$, then
  $\delta_\zeta=m/2+(s_--s_+)/2=m/2+\varepsilon_\zeta$.
\item If $m$ is odd and $c\notin\mathbb R$, then $A_+-A_-=\operatorname{Arg}c-\operatorname{Arg}(-c)=\pi\operatorname{sgn}\operatorname{Im}c$.
  Also $u'=-\operatorname{Im}(c)t^m+\dots$ changes sign, so $\varepsilon_\zeta=\operatorname{sgn}\operatorname{Im}c$ and
  $\delta_\zeta=(m+\varepsilon_\zeta)/2$.
\item If $m$ is odd and $c>0$, then $\delta_\zeta=(m+s_-)/2$. If $m$ is odd and $c<0$, then $\delta_\zeta=(m-s_+)/2$. At an
  extremum both equal $(m+\varepsilon_\zeta)/2$; otherwise they equal $(m+s_\zeta\operatorname{sgn}c)/2$.
\end{itemize}
So $\delta_\zeta=\iota_\zeta$, and (i) follows.

(ii) Write $m=m_\zeta$, $\varepsilon=\varepsilon_\zeta$; recall that $m\le k_\zeta$ and that $k_\zeta$ is odd if and only if
$\varepsilon\neq0$. If $m$ is even and $\varepsilon=\pm1$, then $m\le k_\zeta-1$, so $\iota_\zeta\le m/2+\max(\varepsilon,0)\le(k_\zeta+\varepsilon)/2$.
If $m$ is even and $\varepsilon=0$, then $\iota_\zeta=m/2\le k_\zeta/2$. If $m$ is odd and $\varepsilon\neq0$, the claim is $m\le k_\zeta$. If $m$
is odd and $\varepsilon=0$, then $k_\zeta$ is even, so $m\le k_\zeta-1$ and $\iota_\zeta\le(m+1)/2\le k_\zeta/2$. For an admissible pair
$N_{\mathbb D}(F')=0$; all terms $(k_\zeta+\varepsilon_\zeta)/2$ are non-negative, and $Z_\chi\subset Z_-$, so
$k=\sum_{Z_\chi}\iota_\zeta\le\sum_{Z_\chi}(k_\zeta+\varepsilon_\zeta)/2\le\operatorname{wt}(W)$.
\end{proof}

The term $(k_\zeta+\varepsilon_\zeta)/2$ vanishes exactly at nondegenerate minima ($k_\zeta=1$, $\varepsilon_\zeta=-1$), so
Theorem~\ref{thm:signrule} is the case $\operatorname{wt}(W)=0$ of Theorem~\ref{thm:count}(ii). For Blaschke models ($W$ univalent past
the circle and $F'\neq0$ on the closed disk), Theorem~\ref{thm:count}(i) is the pocket formula, Theorem~\ref{thm:pocket}, in another
form. With base point $0$, the pockets are the points of $Z_-$; a pocket is on if and only if $\chi>0$ there, and off if and only if
$\chi<0$; and the signs of all pockets add up to $0$ (Remark~\ref{rem:countwind}). So $k=\sum_{\text{off}}\varepsilon$, and (ii) is
immediate there. Theorem~\ref{thm:count} adds the zeros of $F'$ on the circle, of any order, together with poles and critical points of $W$ on the circle, and
the bound by $\operatorname{wt}(W)$.

\begin{remark}\label{rem:countwind}
If $W$ is univalent past the circle, then $P=\emptyset$, and $g$ is holomorphic and zero-free near the closed disk, so
$\operatorname{wind}(g|_{\mathbb T})=0$. The proof of Theorem~\ref{thm:count}(i), applied to $g$ in place of $\chi$ (the case $B=1$; the
proof does not use $k\ge1$, and here the special points are exactly the points of $Z_-$ and all have $m_\zeta=0$), gives
$\operatorname{wind}(g|_{\mathbb T})=-\sum_{Z_-}\varepsilon_\zeta$. Hence $\sum_{Z_-}\varepsilon_\zeta=0$ and
$\operatorname{wt}(W)=\frac12\sum_{Z_-}k_\zeta$.
\end{remark}

\begin{corollary}\label{cor:countsharp}
Let $W\in\mathcal S$ be univalent on a neighbourhood of the closed disk, and suppose that $u''\neq0$ at every point of $Z_-$. Then
$\operatorname{wt}(W)$ is the number of points of $Z_-$ at which $u$ has a minimum, and also the number at which it has a maximum. It
is the largest degree of a finite Blaschke product $B$ with $(W,B)$ admissible.
\end{corollary}

\begin{proof}
By Remark~\ref{rem:countwind} with $k_\zeta=1$, $Z_-$ has $n=\operatorname{wt}(W)$ minima and $n$ maxima of $u$. The minima satisfy the
conditions (26) of~\cite[Lemma~8.1]{JatarNg2026b}, so for $n\ge1$ that lemma gives a regular admissible pair of degree $n$; for $n=0$
only $B=1$ is possible. No larger degree is possible, by Theorem~\ref{thm:count}(ii).
\end{proof}

The lower bound in Corollary~\ref{cor:countsharp} is~\cite[Lemma~8.1, Thm.~8.7(b)]{JatarNg2026b}; the upper bound is Theorem~\ref{thm:count}(ii).

\begin{corollary}\label{cor:countpoly}
Let $W\in\mathcal S$, $W(w)\not\equiv w$, be holomorphic on a neighbourhood of the closed disk, and let $N_{u'}$ be the number of zeros of $u'$ on
$\mathbb T$, counted with multiplicity. Then $\deg B\le N_{u'}/2-1$ for every admissible pair $(W,B)$. If $W$ is a polynomial of degree
$n\ge2$, then $N_{u'}\le2n-2$, so $\deg B\le n-2$.
\end{corollary}

\begin{proof}
Let $\zeta^*$ be a point where $u$ is maximal on $\mathbb T$. By the maximum principle $|W|\le|W(\zeta^*)|$ on the closed disk, so
the radial derivative $\operatorname{Re}g=\vartheta'$ is $\ge0$ at $\zeta^*$, and $\zeta^*\notin Z_-$. As $u$ is not constant,
$\zeta^*$ is a strict local maximum, with $(k_{\zeta^*}+1)/2\ge1$. Strict local maxima and minima of $u$ alternate on the circle, so the sum of
$(k_\zeta+\varepsilon_\zeta)/2$ over all zeros of $u'$ is $N_{u'}/2$. Hence $\operatorname{wt}(W)\le N_{u'}/2-1$, and Theorem~\ref{thm:count}(ii)
applies. For a polynomial $W$ of degree $n$, $u'=-\operatorname{Im}(wW'\,\overline W)/|W|^2$ on the circle, and there $wW'\,\overline W$
is a combination of the terms $w^j\bar w^l=e^{i(j-l)t}$ with $1\le j,l\le n$; so $\operatorname{Im}(wW'\,\overline W)$ is a real
trigonometric polynomial of degree at most $n-1$, not identically $0$. So $N_{u'}\le2n-2$.
\end{proof}

For $n=2$ the bound says that a univalent quadratic polynomial carries no non-constant $B$; indeed $w+aw^2$ with $|a|\le1/2$ is
starlike. For the maps
$W=(H\circ\varphi)/N_b$ of~\cite[Lemma~6.2]{JatarNg2026b} where $H$ is a polynomial of degree $n$ with $H(b)=0$, $b\in\mathbb D\setminus\{0\}$,
$\varphi(w)=(w+b)/(1+\bar bw)$ and $N_b=(1-|b|^2)H'(b)$ (Section~\ref{sec:onezero} recalls them), the zeros of $u'$ on the circle
are those of $\operatorname{Im}(zH'\,\overline H)$, in the variable $z=\varphi(w)$ of $H$, moved by $\varphi$ (as $N_b$ is a constant); this trigonometric polynomial has
degree at most $n$, and the same argument gives $\deg B\le n-1$.

\begin{remark}\label{rem:countrecords}
Pairs~\ref{pair:O12F} and~\ref{pair:OT} of Table~\ref{tab:onezero} (Section~\ref{sec:onezero}) are admissible with $F'$ vanishing on the circle, a case that
Theorem~\ref{thm:pocket} does not cover. As in Section~\ref{sec:onezero}, let $z=\varphi(w)$ be the variable of $Q$; then
$\chi=wF'/F=|\varphi'|\,zQ'(z)/Q(z)$ on the circle, where $w\varphi'/\varphi=|\varphi'|>0$, and $\varphi$ preserves the orientation of the circle;
so the indices can be computed in the variable $z$. At $z=1$, the zero of $Q'$ of order $5$ is a maximum of $u$ with $k_\zeta=5$, and $c$ is not real, so the index is $(5+1)/2=3$. The other
points of $Z_\chi$ are two minima of $u$ with $\chi<0$, each of index $-1$; for pair~\ref{pair:OT}, also
the point $\zeta_2\neq1$ of the circle (Table~\ref{tab:onezero}), a simple zero of $Q'$ at a minimum of $u$, of index $0$. So $3-1-1=1=\deg B$. So these two pairs place a zero of order $5$ of $(WB)'$ at a flat maximum of $|W|$ (Section~\ref{sec:flatmax}), where $\arg W$ turns clockwise.
That this zero is a strict maximum of $u$ with $k_\zeta=5$ is proved in Section~\ref{sec:flatmax}.
\end{remark}

\begin{remark}\label{rem:countslits}
Let $W$ be meromorphic past the circle with $W(\mathbb D)=\mathbb C\setminus\Gamma$, as for every extremal $W$ of
Corollary~\ref{cor:Splus}. By the sector count of Lemmas~\ref{lem:D} and~\ref{lem:Dplus}, a finite point of $\Gamma$ is either an end
(one preimage, a zero of $W'$, not in $Z_-$) or lies on an analytic arc of $\Gamma$ (two preimages, on the two edges). At such a point,
$u'=0$ exactly when $\Gamma$ is tangent to the circle about $0$ through it. Then $\vartheta'$ has opposite signs on the two edges, so
exactly one preimage $\zeta$ lies in $Z_-$, with $k_\zeta$ and $\varepsilon_\zeta$ those of the distance to $0$ along $\Gamma$. So $\deg B$ is at most the sum, over these
tangency points, of $(k_\zeta+\varepsilon_\zeta)/2$. A nondegenerate local minimum of the distance to $0$ along $\Gamma$ counts $0$, and a local maximum counts at
least $1$. An extremal pair with $\deg B=k\ge1$ whose $W$ is meromorphic past the circle therefore needs a slit that bends back toward $0$, or touches a circle about $0$ to
higher order, with total weight at least $k$. For straight slits the sum is $0$ (Corollary~\ref{cor:straight}). A slit with a corner is
not covered: $W$ is not meromorphic past the circle at the preimages of a corner.
\end{remark}