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\title{The M\"obius pseudodistance of the annulus and an improved lower bound for Korenblum's constant}
\author{David Niedbala Giraudin\\[2pt]
\small ORCID \href{https://orcid.org/0009-0009-1526-1178}{0009-0009-1526-1178}\\[2pt]
\small Technical assistance: Claude (Anthropic)}
\date{21 October 2026; English version 1 November 2026}

\begin{document}
\maketitle

\begin{abstract}
We show that Korenblum's constant of the Bergman space satisfies $c_2\ge0.446$. The best published
lower bound was $c_2\ge0.4263$ (Wikstr\"om, 2026). The new ingredient is a closed form, in
Jacobi theta functions, of the M\"obius pseudodistance of the annulus $\{c<|z|<1\}$ between two points
of the same centred circle. It separates the variables and yields three monotonicity properties, one of
which, a profile inequality, combined with Kirszbraun's theorem in the hyperbolic plane, improves the
constant in Wang's annular estimate. Wikstr\"om's dual moment criterion is then verified at
$c=223/500$ for an explicit measure with fourteen rational atoms, in rigorous arithmetic, by a
self-contained verifier.
\end{abstract}

\section{Result}

Let $A^2(\D)$ be the Bergman space of the unit disc. Korenblum's maximum principle \cite{Kor,Hay}
asserts the existence of a constant $c\in(0,1)$ such that, for $f,g\in A^2(\D)$, the inequality
$|f(z)|\le|g(z)|$ on $c<|z|<1$ implies $\|f\|_{A^2}\le\|g\|_{A^2}$. The supremum $c_2$ of the
admissible constants is Korenblum's constant.

\begin{theorem}\label{thm:main}
$c_2\ge223/500=0.446$.
\end{theorem}

\begin{center}
\begin{tabular}{lr}
\toprule
Reference & $c_2\ge$\\
\midrule
Hayman 1999 \cite{Hay} & $0.04$\\
Wang 2025 \cite{Wang25} & $0.3554$\\
Wikstr\"om 2026 \cite{Wik} & $0.4263$\\
Pinho, 19 October 2026, unrefereed \cite{Pin} & $0.432$\\
This note, 21 October 2026 & $0.446$\\
\bottomrule
\end{tabular}
\end{center}

\noindent The best known upper bound is $c_2<0.6778994$ (Wang \cite{Wang08}).
The French version of this note is dated 21 October 2026. A pull request to the registry of
optimization constants (\#274, opened in October 2026 by the GitHub user \texttt{vidigoat})
announces $c_2\ge0.446$ by a different argument (a flux lemma and a half-diameter enclosing lemma)
\cite{PR274}; it was not submitted by the author of the present note. R.~Pinho has since obtained
$c_2\ge0.44667$ (hexagon:2610.00696, 31 October 2026, unrefereed) \cite{Pin2}, building on that pull
request, which his note attributes to the present author.

\paragraph{Notation.} Fix $0<c<1$ and the annulus $A_c=\{c<|z|<1\}$. On $\D$,
$\psi(z,w)=|z-w|/|1-\bar wz|$ is the pseudo-hyperbolic distance and $d=2\artanh\psi$ the hyperbolic
distance (curvature $-1$), so that $\sinh(d/2)=\psi/\sqrt{1-\psi^2}$. The M\"obius pseudodistance of
the annulus is $c^*(z_1,z_2)=\sup\{\psi(h(z_1),h(z_2))\}$, the supremum being over holomorphic maps
$h:A_c\to\D$. For $c<r<1$ and $0\le\theta\le2\pi$ we write $\delta_r(\theta)=c^*(r,re^{i\theta})$ and
$D_r(\theta)=2\artanh\delta_r(\theta)$. The theta functions $\vartheta_1,\dots,\vartheta_4$ have nome
$c$ (conventions of \cite[\S20.2]{DLMF}).

\section{Closed form of the pseudodistance on a circle}

Schuster \cite{Sch} gives $\delta_r(\theta)$ as an infinite product; the constant $K_c(r)$ of
\cite[(1)]{Wik} is its value at $\theta=\pi$.

\begin{theorem}\label{thm:closed}
For $c<r<1$, $0<\theta<2\pi$, with $v=\theta/2$ and $y=\log(1/r)$,
\[
\sinh\frac{D_r(\theta)}2=\frac{\vartheta_3(iy)}{|\vartheta_1(iy)|}\cdot\frac{\vartheta_1(v)}{\vartheta_3(v)}.
\]
Equivalently, if $k=(\vartheta_2(0)/\vartheta_3(0))^2$, $k'=(\vartheta_4(0)/\vartheta_3(0))^2$ and
$\Kb=\frac\pi2\vartheta_3(0)^2$ (the complete elliptic integral of modulus $k$, so that
$c=e^{-\pi\Kb'/\Kb}$),
\[
\sinh\frac{D_r(\theta)}2=\frac{\sd(\Kb\theta/\pi,\,k)}{\sd\big(\frac{\Kb}{\pi}\log(1/r^2),\,k'\big)}.
\]
In particular, with $t=r^2$ and $S(t)=\sinh(D_r(\pi)/2)=K_c(r)/\sqrt{1-K_c(r)^2}$,
\[
S(t)=\frac{2\sqrt t}{1-t}\prod_{n\ge1}\frac{(1+c^{2n})^2}{(1-c^{2n-1})^2}\cdot
\frac{(1+c^{2n-1}t)(1+c^{2n-1}/t)}{(1-c^{2n}t)(1-c^{2n}/t)},
\]
a product all of whose factors are $>0$ for $c^2<t<1$.
\end{theorem}

\begin{proof}
Schuster's product reads
$\delta_r(\theta)=\frac cr\big(1+\frac{r^2}c\big)\,2\sin\frac\theta2\prod_{n\ge1}f_ng_n(\theta)$, with
\[
f_n=\frac{(1+r^2c^{2n-1})(1+r^{-2}c^{2n+1})}{(1+c^{2n-1})^2},\qquad
g_n(\theta)=\frac{|1-c^{2n}e^{i\theta}|^2}{|1-r^2c^{2n-2}e^{i\theta}|\,|1-r^{-2}c^{2n}e^{i\theta}|}.
\]
The Jacobi product
\[
\vartheta_1(z)=2c^{1/4}\sin z\prod_{n\ge1}(1-c^{2n})(1-c^{2n}e^{2iz})(1-c^{2n}e^{-2iz}),
\]
taken at $z=v$ and then at $z=v+iy$ (where $e^{2iz}=r^2e^{i\theta}$ and $|\sin z|=|1-r^2e^{i\theta}|/(2r)$),
gives $2\sin\frac\theta2\prod_ng_n(\theta)=\vartheta_1(v)/(r\,|\vartheta_1(v+iy)|)$. Since
\[
\Big(1+\frac c{r^2}\Big)\prod_nf_n=\prod_n\frac{(1+r^2c^{2n-1})(1+r^{-2}c^{2n-1})}{(1+c^{2n-1})^2}
=\frac{\vartheta_3(iy)}{\vartheta_3(0)},
\]
we obtain
\[
\delta_r(\theta)=\frac{\vartheta_3(iy)}{\vartheta_3(0)}\cdot\frac{\vartheta_1(v)}{|\vartheta_1(v+iy)|}.
\]
The addition formula
$\vartheta_3(0)^2\vartheta_1(u+v)\vartheta_1(u-v)=\vartheta_1(u)^2\vartheta_3(v)^2-\vartheta_3(u)^2\vartheta_1(v)^2$
\cite[\S20.7]{DLMF}, taken at $u=iy$, where $\vartheta_1(iy-v)=-\overline{\vartheta_1(v+iy)}$ and
$\vartheta_1(iy)^2=-|\vartheta_1(iy)|^2$, gives
$\vartheta_3(0)^2|\vartheta_1(v+iy)|^2=\vartheta_3(iy)^2\vartheta_1(v)^2+|\vartheta_1(iy)|^2\vartheta_3(v)^2$,
whence
\[
\frac1{\delta_r(\theta)^2}-1=\frac{|\vartheta_1(iy)|^2}{\vartheta_3(iy)^2}\cdot\frac{\vartheta_3(v)^2}{\vartheta_1(v)^2},
\]
which is the first formula. The second follows from
$\vartheta_1(z)/\vartheta_3(z)=\frac{\vartheta_2(0)\vartheta_4(0)}{\vartheta_3(0)^2}\sd\big(\vartheta_3(0)^2z,k\big)$
\cite[\S22.2]{DLMF} and the imaginary transformation $\sd(iu,k)=i\sd(u,k')$ \cite[\S22.6]{DLMF}, with
$\vartheta_3(0)^2y=\frac{\Kb}{\pi}\log(1/r^2)$ and $\sd(\Kb,k)=1/k'$. The third is the first at $v=\pi/2$,
written with the products of $\vartheta_1,\dots,\vartheta_4$.
\end{proof}

\begin{theorem}\label{thm:mono}
Let $c<r<1$.
\begin{enumerate}
\item[(i)] $\theta\mapsto\delta_r(\theta)$ is strictly increasing on $[0,\pi]$; in particular
$\max_\theta\delta_r(\theta)=\delta_r(\pi)=K_c(r)$.
\item[(ii)] For $0\le\theta\le2\pi$, $\sinh\frac{D_r(\theta)}2\le\sin\frac\theta2\cdot\sinh\frac{D_r(\pi)}2$.
\item[(iii)] $t\mapsto S(t)$ is strictly increasing on $[c,1)$, and $S(c^2/t)=S(t)$.
\end{enumerate}
\end{theorem}

\begin{proof}
By the Jacobi products, for $0<v\le\pi/2$,
$\frac{\vartheta_1(v)}{\sin v}=2c^{1/4}\prod_n(1-c^{2n})(1-2c^{2n}\cos2v+c^{4n})$ is increasing in $v$,
and $\vartheta_3(v)=\prod_n(1-c^{2n})(1+2c^{2n-1}\cos2v+c^{4n-2})$ is decreasing. Hence
$\vartheta_1(v)/\vartheta_3(v)$ is increasing, which gives (i) by Theorem~\ref{thm:closed}, and
$\vartheta_1(v)/(\vartheta_3(v)\sin v)$ is maximal at $v=\pi/2$, which gives (ii); the case
$\pi\le\theta\le2\pi$ follows by $\theta\mapsto2\pi-\theta$. For (iii), if $c\le t<1$, the argument
$u'=\frac{\Kb}\pi\log(1/t)$ ranges over $(0,\Kb']$, since $\log(1/c)=\pi\Kb'/\Kb$; on this interval
$\sd(\cdot,k')$ is strictly increasing (its derivative is $\cn/\dn^2>0$), so $S(t)=1/(k'\sd(u',k'))$ is
strictly increasing in $t$. The symmetry comes from the automorphism $z\mapsto c/z$ of $A_c$.
\end{proof}

\begin{remark}
Point (i) is what makes $K_c(r)$ the diameter, for $c^*$, of the circle $|z|=r$, a quantity that
enters the annular estimates of \cite{Wang25,Wik}. Schuster \cite{Sch} publishes a truncated version
of it; Theorem~\ref{thm:mono} gives a complete proof.
\end{remark}

\section{An improved annular estimate}

\begin{proposition}\label{prop:centre}
Let $\omega:A_c\to\D$ be holomorphic, $c<r<1$, and $\rho_0=D_r(\pi)/2=\artanh K_c(r)$. There is
$p\in\D$ such that $d\big(\omega(re^{is}),p\big)\le\rho_0$ for every $s$.
\end{proposition}

\begin{proof}
By the definition of $c^*$ and the rotation invariance of $A_c$,
$d\big(\omega(re^{is}),\omega(re^{is'})\big)\le D_r(|s-s'|)$. On a hyperbolic circle of centre $o$ and
radius $\rho_0$, two points with central angle $\phi$ are at distance $d$ with
$\sinh(d/2)=\sinh\rho_0\sin(\phi/2)$. By Theorem~\ref{thm:mono}(ii), the map sending the point of
angle $s$ of this circle to $\omega(re^{is})$ is therefore $1$-Lipschitz for the hyperbolic distance.
By Kirszbraun's theorem in the hyperbolic plane (Valentine \cite{Val}; see also Lang and Schroeder
\cite{LS}), it extends to a $1$-Lipschitz map defined also at $o$; the image $p$ of $o$ works.
\end{proof}

\begin{proposition}\label{prop:annular}
Let $f(z)=\sum a_kz^k$ and $g(z)=\sum b_kz^k$ be holomorphic in $\D$ with $|f|\le|g|$ on $A_c$. For
$c<r<1$ put $Y(r^2)=\sum_{k\ge1,\ |a_k|<|b_k|}(|b_k|^2-|a_k|^2)r^{2k}$ and
$M_h(r)^2=\frac1{2\pi}\int_0^{2\pi}|h(re^{i\theta})|^2\,d\theta$. Then
\[
Y(r^2)\le G_c(r)\big(M_g(r)^2-M_f(r)^2\big),\qquad
G_c(r)=\frac{1+\sqrt{1+S(r^2)^2}}2=\frac12\Big(1+\frac1{\sqrt{1-K_c(r)^2}}\Big).
\]
\end{proposition}

\begin{proof}
We follow Wang's argument \cite{Wang25} (see \cite[\S2]{Wik}). The quotient $\omega=f/g$ is
holomorphic on $A_c$ (removable singularities) and $|\omega|\le1$; if $|\omega|$ attains $1$, then
$\omega$ is a unimodular constant, $|f|=|g|$ everywhere and the statement is immediate. Otherwise
$\omega$ takes values in $\D$. For complex $\alpha\in\D$, the identity
$|b-\alpha a|^2-|a-\bar\alpha b|^2=(1-|\alpha|^2)(|b|^2-|a|^2)$ and Parseval's identity give
$Y(r^2)\le\frac1{2\pi}\int_0^{2\pi}\frac{|g-\alpha f|^2}{1-|\alpha|^2}(re^{i\theta})\,d\theta$. Now
$\frac{|1-\alpha\omega|^2}{1-|\alpha|^2}=\frac{1-|\omega|^2}{1-\psi(\omega,\bar\alpha)^2}$. With
$\alpha=\bar p$, $p$ given by Proposition~\ref{prop:centre}, we have $\psi(\omega,p)\le\tanh(\rho_0/2)$
on $|z|=r$, hence $|g-\alpha f|^2/(1-|\alpha|^2)\le(|g|^2-|f|^2)\cosh^2(\rho_0/2)$; and
$\cosh^2(\rho_0/2)=\frac{1+\cosh\rho_0}2=G_c(r)$ since $\tanh\rho_0=K_c(r)$.
\end{proof}

\noindent Wang's constant is
$\frac12\big(1+\sqrt{(1+3K^2)/(1-K^2)}\big)=\frac12\big(1+\sqrt{1+4S^2}\big)$, which is strictly larger.

\section{The certificate}

Wikstr\"om \cite[Lemma~3.1]{Wik} shows that $c$ is admissible as soon as there is a probability
measure $\mu$ on $[c^2,1]$, $\mu(\{1\})=0$, with $\int t^k\,d\mu\ge\frac1{k+1}$ for $k\ge0$ and
$\int\beta_c(\sqrt t)\,t^k\,d\mu\le\frac1{k+1}$ for $k\ge1$, where $\beta_c=1-1/G_c$ and $\beta_c(c)=1$.
His proof uses the annular estimate only through the inequality of Proposition~\ref{prop:annular}; it
therefore applies verbatim with our $G_c$.

Take $c=223/500$, $R=99/100$ and $\mu=\sum_{j=1}^{14}w_j\delta_{t_j}+\mathbf 1_{[R,1]}(t)\,dt$, with the
following exact decimal numbers ($t_1=c^2$, $t_{14}=R$, $\sum w_j=R$, so $\mu$ is a probability measure).

\begin{center}
\begin{tabular}{rll}
\toprule
$j$ & $t_j$ & $w_j$\\
\midrule
1 & 0.198916 & 0.42261186256165887\\
2 & 0.60895666661691 & 0.12853389969171364\\
3 & 0.61180781079179 & 0.20469255809899036\\
4 & 0.8374998912126 & 0.0123813743289724\\
5 & 0.83868469913294 & 0.13096698358770436\\
6 & 0.93197625792065 & 0.04984739013621153\\
7 & 0.93247222724023 & 0.00720134704444788\\
8 & 0.96936285312632 & 0.0107784329378477\\
9 & 0.9695862322646 & 0.01227418071154803\\
10 & 0.98448736353123 & 0.00816617890945166\\
11 & 0.98482420908651 & 0.00072802785539435\\
12 & 0.98939709796742 & 0.00144706003927696\\
13 & 0.98955114839119 & 0.00003421625939472\\
14 & 0.99 & 0.00033648783738754\\
\bottomrule
\end{tabular}
\end{center}

\begin{theorem}\label{thm:cert}
For this measure, $A_k=\sum_jw_jt_j^k-\frac{R^{k+1}}{k+1}>0$ and
$B_k=\frac{R^{k+1}}{k+1}+\int_R^1\frac{t^k}{G_c(\sqrt t)}\,dt-\sum_jw_j\beta_c(\sqrt{t_j})\,t_j^k>0$ for
every $k\ge1$; for $k=0$ the first condition is the equality $\mu([c^2,1])=1$.
\end{theorem}

\begin{proof}
Wikstr\"om's two conditions read $A_k\ge0$ and $B_k\ge0$. They are checked directly for
$1\le k\le2943$ (family $A$) and $1\le k\le1400$ (family $B$), with upper bounds for $S$ obtained from
the product of Theorem~\ref{thm:closed} (the remainder of the product being bounded explicitly) and, on
$[R,0.99999]$, a grid of $34\,536$ pieces on which $1/G_c$ is bounded below by its value at the right
endpoint (Theorem~\ref{thm:mono}(iii)) and $\int t^k$ by the midpoint rule (convexity). Beyond these
ranges: $w_{14}(k+1)\ge R$ for $k\ge2943$, hence $w_{14}t_{14}^k\ge R^{k+1}/(k+1)$; and, with
$[a,b]\subset[0.998,0.999]$ and $m=\min_{[a,b]}1/G_c$, the inequality $m(b-a)a^k>\sum_jw_jt_j^k$, true at
$k=1400$, propagates to every $k\ge1400$ because $a>R\ge t_j$, and implies $B_k>0$. The smallest
relative margins are $6.93\cdot10^{-6}$ for $A_k(k+1)/R^{k+1}$ (at $k=407$) and $2.43\cdot10^{-6}$ for
$B_k(k+1)/R^{k+1}$ (at $k=652$).
\end{proof}

\noindent Theorems~\ref{thm:cert} and~\ref{thm:mono}, Proposition~\ref{prop:annular} and
\cite[Lemma~3.1]{Wik} give Theorem~\ref{thm:main}.

\section{Status}

\begin{itemize}
\item \textbf{PROVED}: Theorems~\ref{thm:closed} and~\ref{thm:mono}, Propositions~\ref{prop:centre}
and~\ref{prop:annular}, under the cited results (Schuster's product for $c^*$, classical formulas for
theta and elliptic functions, the Kirszbraun--Valentine theorem, Wikstr\"om's dual lemma).
\item \textbf{VERIFIED} in rigorous arithmetic: Theorem~\ref{thm:cert}, hence Theorem~\ref{thm:main}.
The ancillary verifier gives the margins above; a second, independently written program in ball
arithmetic (Arb), not included here, gives the same margins. The identities of
Theorem~\ref{thm:closed} were also checked numerically to more than 40 digits.
\item \textbf{OPEN}: the value of $c_2$, between $0.446$ and $0.6778994$. The result has not been
reviewed by anyone else.
\end{itemize}

\section{Verification}

The file \texttt{anc/VERIF\_KORENBLUM\_0446.py} (Python~3 with numpy, no argument and no external file,
a few seconds) contains the exact data of the table. It computes upper bounds for $S$ in exact integers
with directed rounding, the moment sums in IEEE floating point with an explicit error bound, calibrates
itself on the seven upper bounds for $\beta_c$ published in \cite[\S5]{Wik}, and then checks $A_k$ and
$B_k$ as above.

\section*{Credits}
The maximum principle is due to Korenblum \cite{Kor} and Hayman \cite{Hay}; the product for the
M\"obius pseudodistance of the annulus to Schuster \cite{Sch}; the annular estimate by splitting of
the coefficients to Wang \cite{Wang25}; the dual moment criterion, and the form of the measures used,
to Wikstr\"om \cite{Wik}. The extension theorem is that of Kirszbraun and Valentine \cite{Val}. The
computations were carried out with the assistance of Claude (Anthropic).

\begin{thebibliography}{10}
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\url{https://dlmf.nist.gov}.
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195--205.
\bibitem{Kor} B.~Korenblum, \emph{A maximum principle for the Bergman space}, Publ.\ Mat.\ \textbf{35}
(1991), 479--486.
\bibitem{LS} U.~Lang, V.~Schroeder, \emph{Kirszbraun's theorem and metric spaces of bounded curvature},
Geom.\ Funct.\ Anal.\ \textbf{7} (1997), 535--560.
\bibitem{Pin} R.~Pinho, \emph{Korenblum's constant: a certificate for $C_{68}\ge0.432$},
hexagon:2610.00399v1, \url{https://github.com/rpinho/korenblum-c68-certificate}.
\bibitem{Pin2} R.~Pinho, \emph{A contact and coefficient correction for Korenblum's constant},
hexagon:2610.00696v1 (31 October 2026), \url{https://hexagonmath.org/2610.00696v1}.
\bibitem{PR274} \texttt{vidigoat}, pull request \#274 to \texttt{teorth/optimizationproblems},
\url{https://github.com/teorth/optimizationproblems/pull/274}.
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pseudodistance for the annulus}, Proc.\ Amer.\ Math.\ Soc.\ \textbf{134} (2006), no.~12, 3525--3530,
doi:10.1090/S0002-9939-06-08378-X.
\bibitem{Val} F.~A.~Valentine, \emph{Contractions in non-Euclidean spaces}, Bull.\ Amer.\ Math.\ Soc.\
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\bibitem{Wang08} C.~Wang, \emph{Domination in the Bergman space and Korenblum's constant}, Integral
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\bibitem{Wang25} C.~Wang, \emph{A lower bound on Korenblum's constant}, Proc.\ Amer.\ Math.\ Soc.\
\textbf{153} (2025), 1209--1214.
\bibitem{Wik} F.~Wikstr\"om, \emph{Moment duality and an improved lower bound for Korenblum's constant},
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\end{thebibliography}

\end{document}