\subsection{An upper bound in degree one}\label{sec:B1upper}

For $0<s<1$ let $B_s$ be the factor $B_a$ of Section~\ref{sec:blaschke} with $a=s$, and put
\[
  B_s(w)=\frac{w-s}{1-sw},\qquad \sigma_s(w)=\frac{wB_s'(w)}{B_s(w)}=\frac{(1-s^2)w}{(w-s)(1-sw)},\qquad L(r)=\log\frac{1+r}{1-r},
\]
and for $\xi\neq0$
write $|\operatorname{Log}\xi|=(\log^2|\xi|+\operatorname{Arg}^2\xi)^{1/2}$, with the principal logarithm $\operatorname{Log}$; it is the
least modulus of a logarithm of $\xi$. For $W\in\mathcal S$ let $\ell$ be the holomorphic logarithm of $wW'(w)/W(w)$ with $\ell(0)=0$. By
Grunsky's theorem~\cite{Grunsky1932} (see also~\cite[Ch.~3]{Duren1983} and~\cite[(18)]{JatarNg2026b}),
\begin{equation}\label{eq:grunsky}
  |\ell(w)|\le L(|w|)\qquad(w\in\mathbb D).
\end{equation}
Jatar and Ng~\cite[Thm.~7.4]{JatarNg2026b} prove $|a|>\sqrt{5/7}$ for the zero $a$ of every admissible pair of degree one, by a
homotopy on the circle $|w|=|a|$. The same homotopy works on other circles.

\begin{lemma}\label{lem:circlehomotopy}
Let $0<s<1$, $r_0=(1-\sqrt{1-s^2})/s$, and let $r_0<\rho<1$ with $\rho\neq s$. If $|\operatorname{Log}(-\sigma_s(w))|>L(\rho)$ for every $w$ with
$|w|=\rho$, then for every $W\in\mathcal S$ the derivative $(WB_s)'$ has a zero in $\{|w|<\rho\}$.
\end{lemma}

\begin{proof}
For $0\le t\le1$ put $\widehat P_t(w)=(w-s)(1-sw)e^{t\ell(w)}+(1-s^2)w$. Since $wW'/W=e^{\ell}$, one has
$(WB_s)'(w)=W(w)\widehat P_1(w)/(w(1-sw)^2)$, as in the proof of~\cite[Thm.~7.4]{JatarNg2026b}, and $W(w)/w$ has no zero in $\mathbb D$; so it
suffices to find a zero of $\widehat P_1$ in $\{|w|<\rho\}$. Let $|w|=\rho$. Then $(w-s)(1-sw)\neq0$, so $\widehat P_t(w)=0$ would mean
$e^{t\ell(w)}=-\sigma_s(w)$; then $t\ell(w)$ is a logarithm of $-\sigma_s(w)$, and $|\operatorname{Log}(-\sigma_s(w))|\le t|\ell(w)|\le L(\rho)$ by
\eqref{eq:grunsky}, a contradiction. Hence no $\widehat P_t$ vanishes on the circle $|w|=\rho$, and by the argument principle the number of
zeros of $\widehat P_t$ in $\{|w|<\rho\}$ does not depend on $t$. The polynomial $\widehat P_0(w)=-sw^2+2w-s$ has the zeros $r_0$ and $1/r_0$, and
$r_0<\rho<1<1/r_0$; so $\widehat P_1$ has exactly one zero in $\{|w|<\rho\}$.
\end{proof}

The case $\rho=s$ is the one used in~\cite{JatarNg2026b}; there $\sigma_s$ has a pole on the circle, and the point $w=s$ is handled by
$\widehat P_t(s)=s(1-s^2)\neq0$.

\begin{theorem}\label{thm:B1upper}
Let $W\in\mathcal S$ and let $B$ be a Blaschke product of degree one with $B(0)>0$ such that $(WB)'$ has no zero in $\mathbb D$. Then the
zero $a$ of $B$ satisfies $|a|>0.84904$. Consequently
\[
  \mathcal B_1<\frac1{0.84904}=1.1778008\ldots<1.177801 .
\]
\end{theorem}

\begin{proof}
Write $a=se^{i\theta_a}$ with $0<s<1$. The function $\widetilde W(w)=e^{-i\theta_a}W(e^{i\theta_a}w)$ lies in $\mathcal S$, and $B(e^{i\theta_a}w)$
is a unimodular multiple of $B_s(w)$; so $(\widetilde WB_s)'$ has no zero in $\mathbb D$. For $s\le\sqrt{5/7}=0.845154\ldots$ this is
impossible by~\cite[Thm.~7.4]{JatarNg2026b}. For $s\in[0.845,0.849]$ we apply Lemma~\ref{lem:circlehomotopy} with $\rho=0.827$, and for
$s\in[0.849,0.84904]$ with $\rho=0.8271$. In both cases $r_0<\rho<s$, because $r_0$ is increasing in $s$ and $r_0(0.84904)<0.5556$; and
the hypothesis $|\operatorname{Log}(-\sigma_s(w))|>L(\rho)$ on $|w|=\rho$ holds for every $s$ in the range by the computation of
Appendix~\ref{app:computations}. Lemma
\ref{lem:circlehomotopy} then gives a zero of $(\widetilde WB_s)'$ in $\mathbb D$, a contradiction. Hence $|a|>0.84904$, and
$1/B(0)=1/|a|<1/0.84904$. A pair of degree zero has gain $1$. Since $\mathcal B_1$ is attained~\cite[Thm.~8.9]{JatarNg2026b},
$\mathcal B_1<1/0.84904$.
\end{proof}


\begin{remark}\label{rem:valueset}
Lemma~\ref{lem:circlehomotopy} uses only~\eqref{eq:grunsky} at each point of the circle,
and not the relations between the values of $\ell$ at different points. Rotations of the Koebe function have $\ell(w)=2\operatorname{artanh}(e^{it}w)$ with $t$ real, so they attain
equality in~\eqref{eq:grunsky} only where $\ell(w)=\pm L(|w|)$. Compare Popov's joint value region of $\log(W(w)/w)$ and $\log W'(w)$ on $\mathcal S$~\cite{Popov1965} (see the survey~\cite{Prokhorov2019})
and Grunsky's value region of $\log(W(w)/w)$~\cite{Grunsky1932}.
\end{remark}

\begin{remark}\label{rem:circlelimit}
The proof of Lemma~\ref{lem:circlehomotopy} works on any Jordan curve in $\mathbb D$ that surrounds $r_0$ and avoids $s$, with the
hypothesis $|\operatorname{Log}(-\sigma_s(w))|>L(|w|)$ on the curve. Numerically, the largest $s$ for which a circle satisfies this hypothesis is about
$0.849043$ (near $\rho=0.8271$); on the circle $\rho=s$ the corresponding value is about $0.845899$.
So Theorem~\ref{thm:B1upper} is close to the limit of this argument. An admissible pair of degree two with gain greater than $1.177801$
would show $\mathcal B_2>\mathcal B_1$, one of the questions of~\cite[Sect.~9]{JatarNg2026b}.
\end{remark}