The Möbius pseudodistance of the annulus and an improved lower bound for Korenblum's constant

Result

Let $A^2(\mathbb{D})$ be the Bergman space of the unit disc. Korenblum’s maximum principle [3, 2] asserts the existence of a constant $c \in(0,1)$ such that, for $f,g \in A^2(\mathbb{D})$, 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.

Reference	$c_2 \ge$
Hayman 1999 [2]	0.04
Wang 2025 [11]	0.3554
Wikström 2026 [12]	0.4263
Pinho, 19 October 2026, unrefereed [5]	0.432
This note, 21 October 2026	0.446

The best known upper bound is $c_2 < 0.6778994$ (Wang [10]). 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 vidigoat) announces $c_2 \ge0.446$ by a different argument (a flux lemma and a half-diameter enclosing lemma) [7]; 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) [6], building on that pull request, which his note attributes to the present author.

Notation. Fix $0 < c < 1$ and the annulus $A_c = \{c < |z| < 1\}$. On $\mathbb{D}$, $\psi(z,w) = |z-w|/|1-w\overline{z}|$ is the pseudo-hyperbolic distance and $d = 2\operatorname{artanh}\psi$ the hyperbolic distance (curvature $-1$), so that $\sinh(d/2) = \psi/\sqrt{1-\psi^2}$. The Möbius pseudodistance of the annulus is $c^*(z_1,z_2)=\sup\{\psi(h(z_1),h(z_2))\}$, the supremum being over holomorphic maps $h:A_c\to\mathbb{D}$. For $c<r<1$ and $0\le\theta\le2\pi$ we write $\delta_r(\theta)=c^*(r,re^{i\theta})$ and $D_r(\theta)=2\operatorname{artanh}\delta_r(\theta)$. The theta functions $\vartheta_1,\ldots,\vartheta_4$ have nome $c$ (conventions of [1 §20.2]).

Closed form of the pseudodistance on a circle

Schuster [8] gives $\delta_r(\theta)$ as an infinite product; the constant $K_c(r)$ of [12 (1)] is its value at $\theta=\pi$.

Proof. By the Jacobi products, for $0<v\le\pi/2$,

is increasing in $v$, and

is decreasing. Hence $\vartheta_1(v)/\vartheta_3(v)$ is increasing, which gives (i) by Theorem 2, 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 $u'=\frac{K}{\pi}\log(1/t)$ ranges over $(0,K']$, since $\log(1/c)=\pi K'/K$; on this interval $\operatorname{sd}(\cdot,k')$ is strictly increasing (its derivative is $\operatorname{cn}/\operatorname{dn}^2>0$), so $S(t)=1/(k'\operatorname{sd}(u',k'))$ is strictly increasing in $t$. The symmetry comes from the automorphism $z\mapsto c/z$ of $A_c$. □

Remark 4. 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 [11, 12]. Schuster [8] publishes a truncated version of it; Theorem 3 gives a complete proof.

An improved annular estimate

Proof. By the definition of $c^*$ and the rotation invariance of $A_c$, $d(\omega(re^{is}),\omega(re^{is'}))\le D_r(|s-s'|)$. 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 3(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 [9]; see also Lang and Schroeder [4]), it extends to a 1-Lipschitz map defined also at $o$; the image $p$ of $o$ works. □

Proof. We follow Wang’s argument [11] (see [12 §2]). 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 $\mathbb{D}$. For complex $\alpha\in\mathbb{D}$, the identity $|b-\alpha a|^2-|a-\overline{\alpha}b|^2=(1-|\alpha|^2)(|b|^2-|a|^2)$ and Parseval’s identity give $Y(r^2)\le\frac{1}{2\pi}\int_0^{2\pi}\frac{|g-\alpha f|^2}{1-|\alpha|^2}(re^{i\theta})\,d\theta$. Now $\frac{1-|\omega|^2}{1-|\alpha|^2}=\frac{1-|\psi(\omega,\overline{\alpha})|^2}{1-|\alpha|^2}$. With $\alpha=\overline p$, given by Proposition 5, 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 $\tanh\rho_0=K_c(r)$. □

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

The certificate

Wikström [12 Lemma 3.1] shows that $c$ is admissible as soon as there is a probability measure $\mu$ on $[c^2,1]$, $\mu(\{1\})=0$, with $\int\beta_c(\sqrt{t})t^k\,d\mu\ge\frac{1}{k+1}$ for $k\ge0$ and $\int\beta_c(\sqrt{t})t^k\,d\mu\le\frac{1}{k+1}$ 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 6; 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).

$j$	$t_j$	$w_j$
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

Proof. Wikström’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 2 (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 3(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 $k=652).$

Theorems 7 and 3, Proposition 6 and [12 Lemma 3.1] give Theorem 1.

Status

• PROVED: Theorems 2 and 3, Propositions 5 and 6, under the cited results (Schuster’s product for $c^*$, classical formulas for theta and elliptic functions, the Kirszbraun–Valentine theorem, Wikström’s dual lemma).

• VERIFIED in rigorous arithmetic: Theorem 7, hence Theorem 1. 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 2 were also checked numerically to more than 40 digits.

• OPEN: the value of $c_2$, between 0.446 and 0.6778994. The result has not been reviewed by anyone else.

Verification

The file 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 [12 §5], and then checks $A_k$ and $B_k$ as above.

Credits

The maximum principle is due to Korenblum [3] and Hayman [2]; the product for the Möbius pseudodistance of the annulus to Schuster [8]; the annular estimate by splitting of the coefficients to Wang [11]; the dual moment criterion, and the form of the measures used, to Wikström [12]. The extension theorem is that of Kirszbraun and Valentine [9]. The computations were carried out with the assistance of Claude (Anthropic).