\subsection{Flat maxima and a Schwarzian on the circle}\label{sec:flatmax}

Corollary~\ref{cor:countsharp} assumes that $u''\neq0$ at every point of $Z_-$. Here we look at a \emph{flat maximum}\label{term:flatmax}: a point
$\zeta\in Z_-$ at which $u$ has a strict local maximum with $k_\zeta\ge3$. Its term in $\operatorname{wt}(W)$ is $(k_\zeta+1)/2\ge2$.
By the definition of the index, at such a point $\iota_\zeta\ge2$ holds only if $m_\zeta\ge3$, or $m_\zeta=2$ and $c_\zeta<0$; and
$\iota_\zeta=(k_\zeta+1)/2$ holds only if $m_\zeta=k_\zeta$, or $m_\zeta=k_\zeta-1$ and $c_\zeta<0$. We show that one number,
computed from $W$ alone, rules this out when it is negative: then $\iota_\zeta\le1$ for every $B$.

For a positive $C^2$ function $f$ on an arc of $\mathbb T$, with derivatives taken in the angle
($f'(\zeta)=\frac d{dt}f(\zeta e^{it})|_{t=0}$), put
\[
  \Upsilon(f)=\frac{f''}f-\frac32\Bigl(\frac{f'}f\Bigr)^2-\frac{1-f^2}2 .
\]
If $Y$ is a real function of the angle with $Y'=f$, then $\Upsilon(f)=\operatorname{Sch}Y+(Y'^2-1)/2$, where $\operatorname{Sch}Y=Y'''/Y'-\frac32(Y''/Y')^2$ is the
Schwarzian derivative. Let $P_a(e^{it})=(1-|a|^2)/|e^{it}-a|^2$ be the Poisson kernel of $a\in\mathbb D$; on the circle,
$\sigma=wB'/B=\sum_jP_{a_j}$, where the $a_j$ are the zeros of $B$.

\begin{lemma}\label{lem:circschw}
\begin{enumerate}
\item[(a)] \textup{(}Classical; see~\cite{OvsienkoTabachnikov2005}.\textup{)} $\Upsilon(P_a)=0$ on $\mathbb T$ for every $a\in\mathbb D$.
\item[(b)] Let $B$ be a finite Blaschke product of degree $k\ge1$, and $\sigma=wB'/B$ on $\mathbb T$. If $k=1$, then $\Upsilon(\sigma)=0$;
  if $k\ge2$, then $\Upsilon(\sigma)>0$ at every point of $\mathbb T$.
\item[(c)] Let $\varphi$ be an automorphism of $\mathbb D$, and let $\psi$ be a real function with $\varphi(e^{i\theta})=e^{i\psi(\theta)}$.
  If $f>0$ and $\tilde f=(f\circ\psi)\,\psi'$, then $\Upsilon(\tilde f)=\psi'^2\,(\Upsilon(f)\circ\psi)$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) After a rotation, $a=r\in[0,1)$. With $D(t)=1+r^2-2r\cos t$ we have $P_a=(1-r^2)/D$, $P_a'/P_a=-D'/D$ and
$P_a''/P_a=-D''/D+2D'^2/D^2$, so
\[
  D^2\,\Upsilon(P_a)=-DD''+\tfrac12D'^2-\tfrac12\bigl(D^2-(1-r^2)^2\bigr).
\]
Here $D'=2r\sin t$, $D''=2r\cos t$ and $D^2-(1-r^2)^2=4r(r-\cos t)(1-r\cos t)$, and the right side expands to $0$.

(b) On the circle $\sigma=\sum_{j=1}^kf_j$ with $f_j=P_{a_j}>0$. By (a), $f_j''=\frac32f_j'^2/f_j+\frac12f_j(1-f_j^2)$. Hence
\begin{align*}
  \sigma\,\Upsilon(\sigma)&=\sum_jf_j''-\frac32\frac{(\sum_jf_j')^2}\sigma-\frac{\sigma(1-\sigma^2)}2\\
  &=\frac32\Bigl(\sum_j\frac{f_j'^2}{f_j}-\frac{(\sum_jf_j')^2}{\sum_jf_j}\Bigr)+\frac12\Bigl(\bigl(\sum_jf_j\bigr)^3-\sum_jf_j^3\Bigr).
\end{align*}
The first bracket is non-negative by the Cauchy--Schwarz inequality. The second is $0$ if $k=1$ and positive if $k\ge2$.

(c) Let $Y'=f$; then $\tilde f=(Y\circ\psi)'$. The chain rule $\operatorname{Sch}(Y\circ\psi)=\psi'^2\,(\operatorname{Sch}Y\circ\psi)+\operatorname{Sch}\psi$ gives
\[
  \Upsilon(\tilde f)=\psi'^2\,(\operatorname{Sch}Y\circ\psi)+\operatorname{Sch}\psi+\tfrac12\bigl(\psi'^2(Y'\circ\psi)^2-1\bigr)=\psi'^2\,(\Upsilon(f)\circ\psi)+\Upsilon(\psi').
\]
On the circle $\psi'=|\varphi'|$ is the Poisson kernel $P_{\varphi^{-1}(0)}$, so $\Upsilon(\psi')=0$ by (a).
\end{proof}

\begin{remark}\label{rem:flatframe}
Let $H$ be a polynomial with no zero on $\mathbb T$, let $\varphi$ be an automorphism of $\mathbb D$, and let $W=C\,(H\circ\varphi)$ with
a constant $C\neq0$. With $\psi$ as in Lemma~\ref{lem:circschw}(c), $\log|W|=\log|H|\circ\psi+\log|C|$ on the circle, and
$wW'/W=\psi'\,(zH'/H)\circ\varphi$ on the circle, with $\psi'=w\varphi'/\varphi>0$ ($H$ has variable $z=\varphi(w)$,
as in Section~\ref{sec:onezero}). So the zeros of $u'$ for $W$ and for $H$
correspond under $\psi$, with the same orders and types, $\vartheta'$ keeps its sign, and by Lemma~\ref{lem:circschw}(c) the number
$\Upsilon(-\vartheta')$ at corresponding points of $Z_-$ has the same sign. We use this for $W=(H\circ\varphi)/N_b$.
\end{remark}

\begin{corollary}\label{cor:flatindex}
Let $W$, $B$ and $\chi$ be as in Section~\ref{sec:count}, let $\zeta\in Z_\chi$ with $m_\zeta\ge2$, and let $f=-\vartheta'$ near $\zeta$
(so $f>0$ there).
\begin{enumerate}
\item[(i)] If $m_\zeta\ge3$, then $\Upsilon(f)(\zeta)=\Upsilon(\sigma)(\zeta)\ge0$, with equality if and only if $k=1$.
\item[(ii)] If $m_\zeta=2$ and $c_\zeta<0$, then $\Upsilon(f)(\zeta)>0$.
\end{enumerate}
\end{corollary}

\begin{proof}
On $\mathbb T\setminus P$ we have $\operatorname{Re}\chi=\operatorname{Re}g+\sigma=\sigma-f$. As $\chi(\zeta e^{it})=c_\zeta t^{m_\zeta}+
O(t^{m_\zeta+1})$, the function $\sigma-f$ vanishes to order at least $m_\zeta$ at $t=0$. If $m_\zeta\ge3$, then $f$, $f'$, $f''$
agree with $\sigma$, $\sigma'$, $\sigma''$ at $\zeta$, so $\Upsilon(f)(\zeta)=\Upsilon(\sigma)(\zeta)$, and Lemma~\ref{lem:circschw}(b) gives
the rest. If $m_\zeta=2$ and $c_\zeta<0$, then $f=\sigma$, $f'=\sigma'$ and $f''=\sigma''-2c_\zeta$ at $\zeta$, so
$\Upsilon(f)(\zeta)=\Upsilon(\sigma)(\zeta)-2c_\zeta/\sigma(\zeta)>0$.
\end{proof}

In particular, if $\deg B=1$, every zero of $F'$ on the circle of order at least $3$ imposes the equation $\Upsilon(-\vartheta')(\zeta)=0$
on $W$ alone, whatever the type of $\zeta$. Every one-zero pair of Table~\ref{tab:onezero}
(Theorem~\ref{thm:tablepairs}, in Section~\ref{sec:onezero}) whose $Q'$ has a zero of order $5$ on the circle has a zero of $F'$ of
order $5$ there, and so satisfies it. For pairs~\ref{pair:O12F}, \ref{pair:OT} and~\ref{pair:O14}, in the variable $z$ of Remark~\ref{rem:countrecords},
$zQ'/Q=c\,t^5+O(t^6)$ at $z=e^{it}$ with $c=iR(1)/Q(1)$, where $Q'=(z-1)^5R$; and $c_\zeta$ is a positive multiple of $c$. In
exact rational arithmetic over $\mathbb Q(i)$, $\operatorname{Re}(R(1)/Q(1))>0$ for all three pairs, so $\operatorname{Im}c_\zeta>0$:
the zero is at a strict local maximum of $u$ with $k_\zeta=5$.

\begin{proposition}\label{prop:flatmax}
Let $W$ be as in Section~\ref{sec:count}, let $\zeta\in Z_-$ be a point at which $u$ has a strict local maximum with $k_\zeta\ge3$,
and put $f=-\vartheta'$ near $\zeta$.
\begin{enumerate}
\item[(i)] If $\Upsilon(f)(\zeta)<0$, then $\iota_\zeta\le1$ for every finite Blaschke product $B$ with $\zeta\in Z_\chi$, and every
  admissible pair $(W,B)$ satisfies $\deg B\le\operatorname{wt}(W)-(k_\zeta-1)/2<\operatorname{wt}(W)$.
\item[(ii)] If $\Upsilon(f)(\zeta)\le0$, then still $\deg B\le\operatorname{wt}(W)-(k_\zeta-1)/2$ for every admissible pair $(W,B)$.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) At a maximum $\varepsilon_\zeta=1$, and by the definition of the index, $\iota_\zeta\ge2$ forces $m_\zeta\ge3$, or $m_\zeta=2$ and
$c_\zeta<0$. In both cases Corollary~\ref{cor:flatindex} gives $\Upsilon(f)(\zeta)\ge0$. So $\iota_\zeta\le1$. For an admissible pair
$N_{\mathbb D}(F')=0$, and Theorem~\ref{thm:count} gives
\[
  k=\sum_{\xi\in Z_\chi}\iota_\xi\le1+\sum_{\xi\in Z_-\setminus\{\zeta\}}\frac{k_\xi+\varepsilon_\xi}2=\operatorname{wt}(W)-\frac{k_\zeta+1}2+1 ;
\]
if $\zeta\notin Z_\chi$, the same bound holds without the term $1$.

(ii) If $\Upsilon(f)(\zeta)=0$ and $k\ge2$, Corollary~\ref{cor:flatindex} excludes $\iota_\zeta\ge2$ (for $k\ge2$ both of its cases
give $\Upsilon(f)(\zeta)>0$), and the proof of (i) applies. If $k=1$, the bound reads $1\le\operatorname{wt}(W)-(k_\zeta-1)/2$, which holds because $\zeta$ alone
contributes $(k_\zeta+1)/2$ to $\operatorname{wt}(W)$.
\end{proof}

\begin{proposition}\label{prop:flatnotsharp}
The hypothesis $u''\neq0$ on $Z_-$ in Corollary~\ref{cor:countsharp} cannot be dropped. More precisely, there is $W\in\mathcal S$,
univalent on a neighbourhood of the closed disk, of the form $W=(H\circ\varphi)/N_b$, where $H$ is a polynomial of degree $14$, $b\in\mathbb D$, $H(b)=0$,
$\varphi(w)=(w+b)/(1+\bar bw)$ and $N_b=(1-|b|^2)H'(b)$, and a point $\zeta\in Z_-$ at which $u$ has a strict local maximum with $k_\zeta=5$ and $\Upsilon(-\vartheta')(\zeta)<0$. For every
such $W$, every admissible pair $(W,B)$ has $\deg B\le\operatorname{wt}(W)-2$.
\end{proposition}

The proof uses pair~\ref{pair:O14} of Theorem~\ref{thm:tablepairs} and Lemmas~\ref{lem:recipek},~\ref{lem:certify} and~\ref{lem:openuniv}; it is given in Section~\ref{sec:fixeddeg}, after Lemma~\ref{lem:openuniv}.

So at a flat maximum the bound of Theorem~\ref{thm:count}(ii) can lose $(k_\zeta-1)/2$, and if
$\Upsilon(-\vartheta')(\zeta)<0$, this loss occurs for every $B$.