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\hypersetup{pdftitle={Smale's mean value constant (1981) exceeds 1.1559, and other miscellanea}, pdfauthor={Dave Goel}}

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\title[Smale's mean value constant (1981) exceeds 1.1559, and other miscellanea]{Smale's mean value constant (1981) exceeds 1.1559,\\
and other miscellanea}
\author{Dave Goel}
\date{October 24, 2026. Revised: October 28 and November 10, 2026}

\begin{document}

\begin{abstract}
We prove that Smale's mean value constant exceeds $1.1559837$, improving the bound $1.1018$ of Jatar and Ng; after the first version of this paper, the author proved that it equals $4$. Smale proved in 1981
that for every complex polynomial $P$ of degree at least $2$ and every point $z$ with $P'(z)\neq0$, some critical point $\zeta$ of
$P$ satisfies $|P(z)-P(\zeta)|\le K\,|z-\zeta|\,|P'(z)|$ with $K=4$, and he conjectured that $K=1$ suffices; $K^*$ is the best
constant, called Smale's mean value constant. That conjecture was disproved in 2026. Jatar and Ng showed that $K^*$
equals the supremum of $1/B(0)$ over the \emph{admissible pairs} $(W,B)$: $W$ is a normalised univalent map of the unit disk, $B$ is a finite Blaschke product
with $B(0)>0$, and the derivative of the product $WB$ has no zero in the disk. We construct such a pair explicitly, with $B$ of degree two, from a
polynomial of degree $18$ with Gaussian rational coefficients, and we verify the required properties by a computation in exact
arithmetic. Writing $\mathcal B_d$ for this supremum restricted to
$\deg B\le d$, we prove, partly with certified computation, that $1.1363375<\mathcal B_1<1.177801$, $\mathcal B_2<100/61$, and
$\mathcal B_d\le4-0.00597/(d\log d)$ for every $d\ge2$, which improves the bound of Jatar and Ng for $d\ge452$. If a pair attains $\mathcal B_d$ with $d\ge1$, then $W$ does not extend
holomorphically to any neighbourhood of the closed disk and is not rational. If $1/B_n(0)\to4$ along a sequence of admissible pairs $(W_n,B_n)$ (such sequences exist, since $K^*=4$), then after
rotation $W_n$ tends to the Koebe function, as Jatar and Ng observed, and the zeros $a$ of $B_n$, weighted by $\log(1/|a|)$, have an explicit limit distribution. Every admissible pair $(W,B)$ with $W$ meromorphic on a neighbourhood of the closed disk has $\deg B\le\operatorname{wt}(W)$, a weighted count of the points of the circle where $|W|$ is stationary and
$\arg W$ decreases.
\end{abstract}

\subjclass[2020]{Primary 30C10; Secondary 30C15, 30C45, 30J10}
\keywords{Smale's mean value conjecture, univalent functions, Blaschke products}

\maketitle

{\small\noindent\textbf{AI Disclosure.} This work was done overwhelmingly by AI models and supervised by the author. The models were
Anthropic's Claude and OpenAI's ChatGPT. The author takes full responsibility for the paper.\par}

\section{Introduction}\label{sec:intro}
\input{sections/sec1_intro}

\input{sections/sec2_prelim}

\input{sections/sec5_pockets}

\input{sections/sec6_structure}

\section{Ownership}\label{sec:own}
\input{sections/sec8_1_ownership}

\section{Symmetric pairs}\label{sec:symmetric}
\input{sections/sec_rt_symmetric}

\input{sections/sec7_permodel}
\input{sections/sec7_FH_insert}
\input{sections/sec7_ES_insert}
\input{sections/sec7_THMS_insert}
\input{sections/sec7_RATEXT_insert}
\input{sections/sec7_THMM_insert}
\input{sections/sec7_FLATMAX_insert}
\input{sections/sec7_DEG_insert}

\input{sections/sec_onezero_D4b}
\input{sections/sec_U1_upper}
\input{sections/sec_U2_B2}

\input{sections/sec_fixeddeg}

\input{sections/sec10_addzeros}

\input{sections/sec9_open}

\appendix
\input{sections/app_pairK2N18}
\input{sections/app_pairN18}
\input{sections/app_computations}

\section*{Acknowledgments}
The author is immensely grateful to Edward Crane for mentorship, and for feedback and suggestions that greatly improved this work
and its exposition.

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\end{document}