\section{Bounds for each Blaschke model, and extremal pairs}\label{sec:permodel}

We keep the notation of Section~\ref{sec:structure}: $(W,B)$ satisfies (M1) and (M2), $k\ge1$, $A$ is the zero \emph{set} of $B$,
the gain $K_0=1/|B(0)|$ and $r_{\rm in}=\operatorname{dist}(0,\partial\Omega)\ge1/4$. In this section we assume $(W B)'\neq0$ only on the
open disk; then $B(0)=(W B)'(0)\neq0$, so $0\notin A$. Hyperbolic densities $\lambda$ (of curvature $-4$), the punctured disk $\mathbb D^*$ and the function
$\hat c(r)=\ell/\sinh \ell$, $\ell=\log(1/r)$, are as in Theorem~\ref{thm:T1}(i) and its proof.

\subsection{The punctured Schwarz bound}

The following lemma carries out the suggestion of~\cite[Rem.~7.3]{JatarNg2026b} to use the hyperbolic metric of
$\mathbb D\setminus A$; compare~\cite{BeardonMindaNg2002}.

\begin{lemma}\label{lem:P}
$r_{\rm in}K_0\le\lambda_{\mathbb D}(0)/\lambda_{\mathbb D\setminus A}(0)\le\min_{a\in A}\hat c(|a|)<1$. Hence
\[
  K_0\ \le\ 4\,\frac{\lambda_{\mathbb D}(0)}{\lambda_{\mathbb D\setminus A}(0)}\ \le\ 4\min_{a\in A}\hat c(|a|)\ <\ 4 .
\]
\end{lemma}

\begin{proof}
Let $g:D(0,r_{\rm in})\to E_0$ be the inverse of Lemma~\ref{lem:branch} at $b=0$. $\Phi$ vanishes at every $W(a)$, $a\in A$, and
$W(a)\neq0$; as $\Phi$ is injective on $E_0$ and $\Phi(0)=0$, $E_0$ contains no $W(a)$. So $h(u):=W^{-1}(g(r_{\rm in}u))$ maps
$\mathbb D$ into $\mathbb D\setminus A$, with $h(0)=0$ and $|h'(0)|=r_{\rm in}/|B(0)|=r_{\rm in}K_0$. Schwarz--Pick for the
hyperbolic metrics gives $\lambda_{\mathbb D\setminus A}(0)\,|h'(0)|\le\lambda_{\mathbb D}(0)=1$. Since
$\mathbb D\setminus A\subset\mathbb D\setminus\{a\}$, $\lambda_{\mathbb D\setminus A}\ge\lambda_{\mathbb D\setminus\{a\}}$; by the proof of
Theorem~\ref{thm:T1}(i), $\lambda_{\mathbb D\setminus\{a\}}(0)=1/\hat c(|a|)$, and $\hat c(|a|)=\ell/\sinh \ell<1$ with $\ell=\log(1/|a|)$. Finally
$r_{\rm in}\ge1/4$.
\end{proof}

\begin{corollary}\label{cor:P}
\textup{(a)} $\hat c$ increases on $(0,1)$, and $r\,\hat c(r)$ increases from $0$ to $1$; let $\rho_*=0.31088522\ldots$ be the root of
$8r^2\log(1/r)=1-r^2$, i.e.\ of $r\,\hat c(r)=1/4$.

\textup{(b)} \emph{Every zero:} $\min_{a\in A}|a|\ge\rho_*$.

\textup{(c)} \emph{Near $4$:} if $K_0\ge4-\delta$, then every zero satisfies $1-|a|\le\sqrt{3\delta/2}$.
\end{corollary}

\begin{proof}
(a) $\frac d{d\ell}\frac \ell{\sinh \ell}=(\sinh \ell-\ell\cosh \ell)/\sinh^2\ell<0$ and $\ell=\log(1/r)$ decreases in $r$, so $\hat c$ increases. Also
$r\,\hat c(r)=u(r^2)$ with $u(q)=q\log(1/q)/(1-q)$, and $u'(q)=(q-1-\log q)/(1-q)^2>0$, $u(0+)=0$, $u(1-)=1$.
(b) $K_0\ge1/|a|$ for every zero (each factor $1/|a_j|\ge1$) and $K_0\le4\hat c(|a|)$ give $|a|\,\hat c(|a|)\ge1/4$, i.e.\
$|a|\ge\rho_*$.
(c) Put $|a|=1-\epsilon$. Then $1-r^2-2r\log(1/r)=\sum_{n\ge3}2\epsilon^n/(n(n-1))\ge\epsilon^3/3$ and $1-r^2\le2\epsilon$, so
$1-\hat c(1-\epsilon)\ge\epsilon^2/6$. From $4-\delta\le K_0\le4\hat c(|a|)$ we get $1-\hat c(|a|)\le\delta/4$, so $\epsilon^2\le3\delta/2$.
\end{proof}

Lemma~\ref{lem:P} bounds each Blaschke model below $4$; it does not exclude $K^*=4$, since the
zeros may approach the circle along a sequence, and (c) gives a rate at which they must. Indeed $K^*=4$~\cite{GoelK4}.

\begin{remark}
The bounds of~\cite[Thm.~7.1]{JatarNg2026b} are stronger than (b) and (c): there $B(0)\ge(1+|a|)^2/(16|a|)$ for every
zero $a$, so every zero has $|a|\ge1/3$, and $K_0\ge4-\delta$ forces $(1-|a|)^2\le4|a|\delta/(4-\delta)$. Corollary~\ref{cor:FH} and
Theorem~\ref{thm:DEG} need only (b) and (c); the constants of Theorem~\ref{thm:DEG} are computed from (c).
\end{remark}

\begin{lemma}[Smale at every zero]\label{lem:Z}
Every zero $a$ of $B$ is simple, and
\[
  |W(a)|\prod_{b\in A\setminus\{a\}}|B_b(a)|\ \ge\ r_{\rm in}\,(1-|a|^2)\,\lambda_X(a)\ \ge\ r_{\rm in},\qquad
  X:=\mathbb D\setminus(\{0\}\cup(A\setminus\{a\}));
\]
equivalently $\sum_{b\neq a}g_{\mathbb D}(a,b)\le\log(|W(a)|/r_{\rm in})$. Moreover $|W(a)|\ge r_{\rm in}/\hat c(|a|)>r_{\rm in}$.
\end{lemma}

\begin{proof}
$(W B)'(a)=W(a)B'(a)\neq0$, as $(W B)'\neq0$ on $\mathbb D$, so $B'(a)\neq0$ and $W(a)\neq0$. Let $g_a:D(0,r_{\rm in})\to E_{W(a)}$ be the inverse of
Lemma~\ref{lem:branch} at $b=W(a)$, and $f:=W^{-1}\circ g_a$. Then $f(0)=a$ and
$f'(0)=1/(W'(a)\Phi'(W(a)))=1/(W(a)B'(a))$. $E_{W(a)}$ contains no other zero of $\Phi$, so $f$ maps $D(0,r_{\rm in})$ into $X$,
and Schwarz--Pick gives $r_{\rm in}\lambda_X(a)\le1/|f'(0)|=|W(a)||B'(a)|$. Now $(1-|a|^2)|B'(a)|=\prod_{b\neq a}|B_b(a)|$, and
$(1-|a|^2)\lambda_X(a)\ge1$ as $X\subset\mathbb D$. For the last claim use $X\subset\mathbb D^*$: $\lambda_X(a)\ge\lambda_{\mathbb
D^*}(a)=1/(2|a|\log(1/|a|))$, together with $|B'(a)|\le1/(1-|a|^2)$ (Schwarz--Pick for $B$, $B(a)=0$).
\end{proof}

\subsection{Any domain pair}

\begin{proposition}\label{prop:anypair}
Let $(\Omega,\Phi)$ be a domain pair with $\mu=\mu(\Omega,\Phi)>0$, $r_{\rm in}=\operatorname{dist}(0,\partial\Omega)$, and let $Z_\Phi$ be the
\textup{(}finite\textup{)} set of zeros of $\Phi$ in $\Omega$ other than $0$. Then $\mu\,r_{\rm in}\le1/\lambda_{\Omega\setminus {Z_\Phi}}(0)$.
\end{proposition}

\begin{proof}
$Z_\Phi$ is finite, since $\Phi\not\equiv0$ has isolated zeros in the compact set $\overline\Omega$. On $\partial\Omega$,
$|\Phi(z)|\ge\mu|z|\ge\rho:=\mu r_{\rm in}$. The proof of Lemma~\ref{lem:branch}, with $r_{\rm in}$ replaced by $\rho$ and its boundary case replaced by the continuity of
$\Phi$ on $\overline\Omega$ (a limit point $z\in\partial\Omega$ of points of $E$ has $|\Phi(z)|\ge\mu|z|\ge\rho$), gives a component $E$
of $\{z\in\Omega:|\Phi(z)|<\rho\}$ containing $0$ and an inverse $g:D(0,\rho)\to E$ with $g(0)=0$, $g'(0)=1/\Phi'(0)=1$; $E$ misses $Z_\Phi$
because $\Phi$ is injective on $E$. Schwarz--Pick: $\lambda_{\Omega\setminus {Z_\Phi}}(0)\le\lambda_{D(0,\rho)}(0)=1/\rho$.
\end{proof}

\begin{corollary}\label{cor:simplyconnected}
$\mu(\Omega,\Phi)<4$ for every domain pair with $\Omega$ simply connected.
\end{corollary}

\begin{proof}
Let $W:\mathbb D\to\Omega$ be the Riemann map with $W(0)=0$ and $W'(0)=r(\Omega,0)>0$. By conformal invariance
$\lambda_{\Omega\setminus {Z_\Phi}}(0)=\lambda_{\mathbb D\setminus W^{-1}(Z_\Phi)}(0)/r(\Omega,0)$, so Proposition~\ref{prop:anypair} gives
\[
  \mu\ \le\ \frac{r(\Omega,0)}{r_{\rm in}}\cdot\frac{\lambda_{\mathbb D}(0)}{\lambda_{\mathbb D\setminus W^{-1}(Z_\Phi)}(0)},
\]
and $r(\Omega,0)\le4r_{\rm in}$ by Koebe's theorem. If $\mu\le1$ there is nothing to prove. If $\mu>1$, then $Z_\Phi\neq\emptyset$ by
Remark~\ref{rem:zeros}, and $\lambda_{\mathbb D\setminus W^{-1}(Z_\Phi)}(0)\ge1/\hat c(|a|)>1$ for any $a\in W^{-1}(Z_\Phi)$ (proof of
Lemma~\ref{lem:P}); so $\mu<4$.
\end{proof}

For Blaschke models ($r(\Omega,0)=1$, $Z_\Phi=W(A)$, and $\mu=K_0$ after $\Phi\mapsto \Phi/B(0)$) Proposition~\ref{prop:anypair} is
Lemma~\ref{lem:P}.