\subsection{Rational extremals}\label{sec:ratext}

Corollary~\ref{cor:S2} excludes a rational $W$ without poles on the closed disk, and Corollary~\ref{cor:Splus} says that if a
rational $W$ with poles on the circle belongs to a pair that attains $\mathcal B_d$ with $d\ge1$, then $W(\mathbb D)$ is a slit plane. We show
that a rational map onto a slit plane has straight slits (Proposition~\ref{thm:ratslit}), that such a map carries no pair with
non-constant $B$ (Corollary~\ref{cor:straight}, from the sign rule of Theorem~\ref{thm:signrule}), and so that no extremal pair has a rational $W$
(Corollary~\ref{cor:ratext}). Let $\mathcal J(w)=w+1/w$, the Joukowski map; it maps $\mathbb D$ conformally onto the sphere minus
$[-2,2]$. Proposition~\ref{thm:ratslit} below follows from the classical fact that a rational (resp.\ entire) Schwarz function forces
a circle or a line (resp.\ a line); see~\cite{DavisPollak1958} and~\cite[pp.~104--106]{Davis1974}. The part proved here is that
the Schwarz function of a rational univalent $W$ with dense image is a M\"obius map, which gives the form~(iii) of $W$.

\begin{proposition}\label{thm:ratslit}
Let $W$ be a rational function that is univalent in $\mathbb D$, with no pole in $\mathbb D$, such that $W(\mathbb D)$ is dense in
$\mathbb C$. Let $P$ be the set of poles of $W$ on $\mathbb T$, and $\Gamma=\mathbb C\setminus W(\mathbb D)$.
\begin{enumerate}
\item[(i)] $\Gamma=W(\mathbb T\setminus P)$; it is non-empty and closed, with empty interior and of locally finite length.
\item[(ii)] $\Gamma$ lies on a line: it is one closed ray, or two disjoint closed rays on one line that point away from each other.
\item[(iii)] $W=\Sigma\circ\mathcal J\circ\psi$, where $\psi$ is an automorphism of $\mathbb D$ and $\Sigma$ is a M\"obius map whose pole lies
  in $[-2,2]$; in particular $\deg W=2$.
\end{enumerate}
Conversely, every such $\Sigma\circ\mathcal J\circ\psi$ is univalent in $\mathbb D$, and its image is $\mathbb C$ minus one or two closed
rays on a line.
\end{proposition}

The image of a rational $W$ in Corollary~\ref{cor:Splus} is a slit plane, and a slit plane is dense in $\mathbb C$: a finite union of
analytic arcs that is closed in $\mathbb C$ has empty interior. So Proposition~\ref{thm:ratslit} applies to it.

\begin{proof}
Replacing $W$ by $(W-W(0))/W'(0)$, an affine change that keeps the hypotheses and the conclusions, we may assume $W\in\mathcal S$.
As $W(\mathbb D)$ is dense, it is unbounded, so $P\neq\emptyset$. Put $\Omega=W(\mathbb D)$.

(i) Let $z_0\in\Gamma$ and $W(w_n)\to z_0$ with $w_n\in\mathbb D$. A limit point $\zeta$ of $(w_n)$ is not in $\mathbb D$, since
$z_0\notin\Omega$, and not in $P$, since $W\to\infty$ there; so $\zeta\in\mathbb T\setminus P$ and $W(\zeta)=z_0$. Conversely
$W(\mathbb T\setminus P)\cap\Omega=\emptyset$ by Lemma~\ref{lem:poles}. So $\Gamma=W(\mathbb T\setminus P)$. It is closed, as $\Omega$
is open; it is non-empty, since the inverse of a conformal map of $\mathbb D$ onto $\mathbb C$ would be a bounded non-constant entire
function; and it has empty interior, as $\Omega$ is dense. For a compact set $X\subset\mathbb C$, the points $\zeta\in\mathbb
T\setminus P$ with $W(\zeta)\in X$ stay away from $P$ and form a compact subset of $\mathbb T\setminus P$, on which $W$ is analytic;
so $\Gamma\cap X$ has finite length.

\emph{The Schwarz function.} The function $W^*(w)=\overline{W(1/\bar w)}$ is rational, so $S_\Omega=W^*\circ W^{-1}$ is meromorphic on
$\Omega$, with finitely many poles; they lie in the open set $\Omega$, at positive distance from $\Gamma$. Let $z_0\in\Gamma$ and
$z_n\to z_0$ with $z_n\in\Omega$, and put $w_n=W^{-1}(z_n)$. As in (i), every limit point $\zeta$ of $(w_n)$ lies in
$\mathbb T\setminus P$ with $W(\zeta)=z_0$; $W^*$ is continuous at $\zeta$, and $W^*(\zeta)=\overline{W(\zeta)}$ since
$1/\bar\zeta=\zeta$. So $S_\Omega(z_n)\to\overline{z_0}$, whichever preimage the $w_n$ approach, and $S_\Omega$ extends continuously to $\mathbb C$
minus its poles, with $S_\Omega(z)=\bar z$ for $z\in\Gamma$.

\emph{$S_\Omega$ is meromorphic on $\mathbb C$.} $S_\Omega$ is continuous on $\mathbb C$ minus finitely many poles and holomorphic off the closed
set $\Gamma$, which has $\sigma$-finite length by (i). By a theorem of Besicovitch~\cite{Besicovitch1931}, a function that is
continuous on an open set and holomorphic off a closed subset of $\sigma$-finite length is holomorphic there. So $S_\Omega$ is meromorphic
on $\mathbb C$.

\emph{$S_\Omega$ is a M\"obius map.} $S_\Omega$ is not constant, as $S_\Omega(z)=\bar z$ on $\Gamma$, which contains an arc. Put
$S_\Omega^*(\xi)=\overline{S_\Omega(\bar\xi)}$, meromorphic on $\mathbb C$. For $z\in\Gamma$, $\overline{S_\Omega(z)}=z$, so $S_\Omega^*(S_\Omega(z))=\overline{S_\Omega(z)}=z$;
as $\Gamma$ has non-isolated points, $S_\Omega^*\circ {S_\Omega}$ is the identity. So $S_\Omega$ is injective on $\mathbb C$ minus its poles. It has at most
one pole, since near two poles the values near $\infty$ would be taken twice (open mapping theorem). It has no essential
singularity at $\infty$: if $R$ exceeds the moduli of all poles, then by the Casorati--Weierstrass theorem $S_\Omega(\{|z|>R\})$ is dense in
$\mathbb C$, so it meets the non-empty open set $S_\Omega(\{|z|<R\}\setminus\{\text{poles}\})$, against injectivity. So $S_\Omega$ is rational and
injective, that is, a M\"obius map.

(ii) $A(z)=\overline{S_\Omega(z)}$ is an antiholomorphic map of the sphere with $A\circ A=S_\Omega^*\circ {S_\Omega}=\mathrm{id}$ and $A(z)=z$ on $\Gamma$. An
antiholomorphic involution of the sphere is conjugate by a M\"obius map to $z\mapsto\bar z$, whose fixed set is a circle, or to
$z\mapsto-1/\bar z$, which has no fixed point. So $\Gamma$ lies on a circle $C_0$. As $\Omega$ is simply connected, the complement
$\Gamma\cup\{\infty\}$ of $\Omega$ in the sphere is connected; so the non-empty closed set $\Gamma$ is unbounded, and $\infty$ lies in
the closed set $C_0$: $C_0$ is a line together with $\infty$. Now $\Gamma\cup\{\infty\}$ is a connected closed proper subset of $C_0$
(the plane minus a line is not connected) with more than one point, that is, a closed arc of $C_0$ through $\infty$. If $\infty$ is an
end of the arc, $\Gamma$ is one closed ray; otherwise $\Gamma$ is the line minus a bounded open interval, that is, two closed rays.

(iii) A M\"obius map $\Sigma$ maps $[-2,2]$ onto the arc $\Gamma\cup\{\infty\}$ (ends to ends and one inner point to one inner point), so
$\Omega=\Sigma(\text{sphere}\setminus[-2,2])$, and the pole of $\Sigma$ lies in $[-2,2]$. Then $\Sigma\circ\mathcal J$ and $W$ are conformal maps of
$\mathbb D$ onto $\Omega$, so $W=\Sigma\circ\mathcal J\circ\psi$ with $\psi$ an automorphism of $\mathbb D$, and $\deg W=\deg\mathcal J=2$.
The converse is the same computation read backwards.
\end{proof}

\begin{remark}\label{rem:ratslit}
(a) If a rational $W$, univalent in $\mathbb D$ and possibly with a pole there, has $W(\mathbb D)$ dense in the sphere, then the
sphere minus $W(\mathbb D)$ is a closed circular arc and $W=\Sigma\circ\mathcal J\circ\psi$ with $\psi$ an automorphism of $\mathbb D$ and $\Sigma$ a M\"obius map whose pole need not lie in $[-2,2]$: apply
Proposition~\ref{thm:ratslit} to the composition of $W$ with a M\"obius map that sends a point outside $W(\mathbb D)$ to $\infty$.
(b) For one ray (the pole of $\Sigma$ at $\pm2$) $W$ has one double pole on $\mathbb T$, since $\mathcal J(w)-2=(w-1)^2/w$; for two rays (the
pole of $\Sigma$ in $(-2,2)$) it has two simple poles on $\mathbb T$, since $\mathcal J(w)=x$ has two roots on $\mathbb T$ for $-2<x<2$. This
is the alternative of Corollary~\ref{cor:Splus}.
(c) Examples: the Koebe function, $1/(\mathcal J(w)-2)$, equal to $1/(2\cos t-2)$ at $w=e^{it}$, omits the ray $(-\infty,-1/4]$;
$w/(1+w^2)=1/\mathcal J(w)$, equal to $1/(2\cos t)$ on $\mathbb T$, omits $(-\infty,-1/2]\cup[1/2,\infty)$; $w/(1-w^2)$, equal to
$i/(2\sin t)$ on $\mathbb T$, omits two rays on the imaginary axis.
(d) So no rational map of $\mathbb D$ onto a slit plane has a curved slit.
\end{remark}

When all poles of $W$ lie on the circle, the Schwarz function $S_\Omega$ of the proof of Proposition~\ref{thm:ratslit} has no pole; compare Sakai's null quadrature
domains~\cite{Sakai1981}, and Shapiro~\cite{Shapiro1987} for unbounded quadrature domains.

\medskip\noindent\emph{A sign rule.} Let $W\in\mathcal S$ be meromorphic on an open set $U\supset\overline{\mathbb D}$, with its poles
in $U$ forming a finite set $P\subset\mathbb T$, as before Lemma~\ref{lem:poles}. By that lemma $W(\mathbb T\setminus P)$ misses
$W(\mathbb D)\ni0$, so $W\neq0$ on $\mathbb T\setminus P$. Put $g=wW'/W$; it is holomorphic near the closed disk minus $P$ and
zero-free in $\mathbb D$, with $g(0)=1$. On $\mathbb T\setminus P$ write $W(e^{it})=e^{u(t)+i\vartheta(t)}$ locally, so that
\[
  g(e^{it})=\vartheta'(t)-iu'(t),\qquad u=\log|W| .
\]
$u$ is real-analytic on $\mathbb T\setminus P$ and tends to $+\infty$ at $P$, so it is constant on no arc of $\mathbb T\setminus P$,
unless $P=\emptyset$ and $|W|$ is constant on $\mathbb T$; then $W(\mathbb D)$ is a disk about $0$ and $W(w)=w$. Near a pole $p$ of
order $m$, $g=-mp/(w-p)+O(1)$, so along the circle
\[
  g(pe^{it})=\frac{im}{2}\cot\frac t2+O(1)\qquad(t\to0).
\]
Hence the set
\label{term:Zminus}\[
  Z_-=\{\zeta\in\mathbb T\setminus P:\ g(\zeta)\in(-\infty,0)\}
\]
stays away from $P$; it consists of zeros of $u'$, as $\operatorname{Im}g=-u'$, so it is finite. At $\zeta=e^{it}\in Z_-$ the map
$t\mapsto\arg W(e^{it})$ turns clockwise and $|W|$ is stationary. For a pair $(W,B)$ put $F=WB$, $\sigma=wB'/B$, which is holomorphic near
$\mathbb T$ and positive there (the function $\sigma$ of Section~\ref{sec:criterion}), and
\[
  \chi=g+\sigma=\frac{wF'}{F}.
\]

\begin{theorem}\label{thm:signrule}
Let $W$ be as above, and suppose that $u''(t)>0$ at every $e^{it}\in Z_-$; that is, every stationary point of $|W|$ on the circle at
which $\arg W$ turns clockwise is a nondegenerate minimum of $|W|$. Then there is no admissible pair $(W,B)$ with $B$ non-constant.
\end{theorem}

If $W$ is starlike, then $\operatorname{Re}g>0$ in $\mathbb D$, so $Z_-=\emptyset$; for such $W$, meromorphic past the circle,
Theorem~\ref{thm:signrule} recovers the observation of~\cite[Sect.~9]{JatarNg2026b} that a starlike $W$ admits no non-constant
factor $B$.

\begin{proof}
Let $(W,B)$ be admissible with $\deg B=k\ge1$; the zeros $a_1,\dots,a_k$ of $B$ are simple and non-zero (Section~\ref{sec:ES}). On
$\mathbb T\setminus P$, $F\neq0$, so the zeros of $\chi$ there are the zeros of $F'$. At such a point $\zeta=e^{it}$,
$\operatorname{Im}\chi=-u'(t)=0$ and $\operatorname{Re}\chi=\vartheta'(t)+\sigma(\zeta)=0$, so $g(\zeta)=-\sigma(\zeta)<0$ and
$\zeta\in Z_-$; there $\frac{d}{dt}\chi(e^{it})$ has imaginary part $-u''(t)\neq0$, so the zero is simple. At a zero $c$ of $W'$ on
$\mathbb T\setminus P$, $\chi(c)=\sigma(c)>0$.

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
$\mathbb T\setminus P$, where $2\rho$ is less than the distance between any two points of $P\cup Z_-$, and no zero of $B$ lies within
$2\rho$ of $\mathbb T$. Inside $\mathbb T_\rho$, $\chi$ has no zero, as $F'\neq0$ in $\mathbb D$ and $\chi(0)=1$, and it has exactly $k$
poles, all simple, at the $a_j$, as $W\neq0$ in $\mathbb D\setminus\{0\}$. By the argument principle
$\operatorname{wind}(\chi\circ\mathbb T_\rho)=-k<0$. For small $\rho$ the curve $\chi\circ\mathbb T_\rho$ misses $0$ and meets $(-\infty,0)$ at finitely
many points (see below), and its winding number is the sum of the local crossing numbers there: $+1$ for a passage from
$\operatorname{Im}\chi>0$ to $\operatorname{Im}\chi<0$, $-1$ for the opposite passage, and $0$ if the sign does not change.

(a) On the arcs of $\mathbb T$ outside the indents, $\operatorname{Im}\chi=-u'$, as $\sigma$ is real. So $\chi$ meets $(-\infty,0)$ only
at zeros of $u'$ where $\vartheta'+\sigma<0$; there $g=\vartheta'<-\sigma<0$, so the point lies in $Z_-$ and is a nondegenerate minimum
of $u$. So $u'$ passes from negative to positive, and $\operatorname{Im}\chi$ from positive to negative: the crossing number is $+1$.

(b) At an indent about a zero $\zeta$ of $\chi$, put $A_\zeta=i\zeta\chi'(\zeta)$, so that $\chi(\zeta e^{it})=A_\zeta t+O(t^2)$ and
$\operatorname{Im}{A_\zeta}=-u''<0$. On the inward semicircle $w-\zeta$ runs from $-i\zeta\rho$ through $-\zeta\rho$ to $i\zeta\rho$, clockwise
about $\zeta$, so $\chi=\chi'(\zeta)(w-\zeta)+O(\rho^2)$ runs from $-A_\zeta\rho$ through $iA_\zeta\rho$ to $A_\zeta\rho$: it starts in the upper
half-plane, passes through the right half-plane ($\operatorname{Re}(iA_\zeta)=-\operatorname{Im}{A_\zeta}>0$) and ends in the lower half-plane. It
meets the positive axis, not $(-\infty,0)$: the indent contributes $0$.

(c) At an indent about a pole $p$ of order $m$, write $w-p=\rho pe^{i\phi}$ with $\phi\in[\pi/2,3\pi/2]$. As $\sigma$ is bounded near
$p$, $\chi=-(m/\rho)e^{-i\phi}+O(1)$, so $\arg\chi=\pi-\phi+O(\rho)$ lies in $[-\pi/2,\pi/2]$ up to $O(\rho)$, and where
$\operatorname{Im}\chi$ is small, $\phi$ is near $\pi$ and $\operatorname{Re}\chi$ is near $m/\rho>0$. The indent does not meet
$(-\infty,0]$: it contributes $0$.

So $\operatorname{wind}(\chi\circ\mathbb T_\rho)$ is the number of crossings in (a), which is at least $0$, against $-k<0$.
\end{proof}

\begin{remark}\label{rem:signrule}
For Blaschke models, Theorem~\ref{thm:signrule} is contained in the pocket formula: with base point $0$, $v=1/g$, so the pockets
are the points of $Z_-$, a minimum of $u$ is a pocket of sign $-1$ and a maximum one of sign $+1$; and by
Proposition~\ref{prop:guard} a feasible triple has a pocket of sign $+1$. The method, the argument principle for $wF'/F$ along the
circle, is that of~\cite[Sect.~8]{JatarNg2026b} and of Proposition~\ref{prop:criterion}. What Theorem~\ref{thm:signrule} adds is the
treatment of the boundary: indents at poles of $W$ on the circle and at zeros of $(WB)'$ there; at critical points of $W$ on the
circle (for instance the ends of slits) $\chi=\sigma>0$, so no indent is needed. This does
not contradict~\cite[Lemma~8.1, Thm.~8.7(b)]{JatarNg2026b}, which build admissible pairs with non-constant $B$ from nondegenerate minima in $Z_-$:
if $W$ has no pole or critical point on the circle, for instance if it is univalent past the circle, then $g$ is holomorphic and zero-free near the closed disk, $\operatorname{wind}(g)=0$, and
under the hypothesis of Theorem~\ref{thm:signrule} each point of $Z_-$ adds $+1$ to this winding number, so $Z_-=\emptyset$. So
the theorem has content only when $W$ has poles or critical points on the circle.
\end{remark}

\begin{corollary}\label{cor:straight}
Let $W=\Sigma\circ\mathcal J\circ\psi\in\mathcal S$ be as in Proposition~\ref{thm:ratslit}\textup{(iii)}, so that $W(\mathbb D)$ is $\mathbb C$
minus one closed ray, or minus two disjoint closed rays on one line. Then there is no admissible pair $(W,B)$ with $B$ non-constant.
\end{corollary}

\begin{proof}
$W$ is rational, so the setting above applies, and $\Gamma=W(\mathbb T\setminus P)$ by Proposition~\ref{thm:ratslit}(i). At a zero of
$W'$ on $\mathbb T\setminus P$ (an end of a ray) $g=0$, so it is not in $Z_-$. At the other points of $\mathbb T\setminus P$, the map
$t\mapsto W(e^{it})$ is a local diffeomorphism onto a ray $\{z_e+se^{i\theta}:s\ge0\}$, so locally $W(e^{it})=z_e+s(t)e^{i\theta}$
with $s'(t)\neq0$. The function $s\mapsto|z_e+se^{i\theta}|^2$ is a strictly convex quadratic, positive as $0\notin\Gamma$, so
$\ell(s)=\log|z_e+se^{i\theta}|$ has at most one stationary point, and there $\ell''=1/|z_e+se^{i\theta}|^2>0$. As $u=\ell\circ s$, a zero of $u'$ is
a stationary point of $\ell$, and there $u''=\ell''\,(s')^2>0$. So every point of $Z_-$ is a nondegenerate minimum of $u$, and
Theorem~\ref{thm:signrule} applies.
\end{proof}

Corollary~\ref{cor:straight} needs straight slits: Jatar and Ng~\cite[Sect.~9]{JatarNg2026b} mention a pair with $\deg B=1$ and gain $\sqrt{61/58}$, built from a bent
radial slit.

\begin{corollary}\label{cor:ratext}
Let $d\ge1$. No pair that attains $\mathcal B_d$ has a rational $W$.
\end{corollary}

\begin{proof}
Let $(W,B)$ attain $\mathcal B_d$ with $W$ rational. $B$ is not constant, as $1/B(0)=\mathcal B_d\ge\mathcal B_1>1$
by~\cite[Thm.~6.1]{JatarNg2026b}. $W\in\mathcal S$ is rational, so it is meromorphic on $\mathbb C$, and by
Corollary~\ref{cor:Splus}, $W(\mathbb D)$ is a slit plane, which is dense in $\mathbb C$. By Proposition~\ref{thm:ratslit},
$W=\Sigma\circ\mathcal J\circ\psi$ with the pole of $\Sigma$ in $[-2,2]$, and Corollary~\ref{cor:straight} gives a contradiction.
\end{proof}

So an extremal $W$, which exists for each $d\ge1$ by~\cite[Thm.~8.9]{JatarNg2026b}, is not rational. By Corollaries~\ref{cor:S}
and~\ref{cor:Splus} it does not extend holomorphically past the circle, and if it extends meromorphically, then it is a
non-rational map onto a slit plane; by Theorem~\ref{thm:signrule}, $Z_-$ then has a point at which $u$ is not a nondegenerate
minimum.