% Copyright 2026 Ricardo Pinho. CC BY 4.0. \documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb} \usepackage[expansion=false]{microtype} \usepackage[hyphens]{url} \usepackage[hidelinks]{hyperref} \setlength{\emergencystretch}{2em} \newtheorem{theorem}{Theorem} \title[A contact correction for Korenblum's constant]{A contact and coefficient correction for Korenblum's constant} \author{Ricardo Pinho} \address{Independent researcher} \thanks{ORCID: \href{https://orcid.org/0009-0006-2744-2599}{0009-0006-2744-2599}. Licensed under CC BY 4.0.} \date{30 October 2026; unrefereed preprint} \subjclass[2020]{30H20, 30C80, 30C85} \keywords{Korenblum's maximum principle, Bergman space, annular Green function, contact moments, outer functions, interval certificates} \hypersetup{pdftitle={A contact and coefficient correction for Korenblum's constant},pdfauthor={Ricardo Pinho}} \begin{document} \begin{abstract} Retaining an angular deficit in an annular coefficient estimate gives $C_{68}\ge0.44667$ for Korenblum's constant in the unweighted Bergman space. The flux input is David Niedbala Giraudin's. A contact-moment estimate and an even outer factor retain a correction involving the first two coefficients; a rational measure then satisfies the modified moment conditions. The argument builds on Wang's coefficient approach and Wikstr\"om's moment integration. This note is unrefereed, with no independent human review or formalization. \end{abstract} \maketitle \section{Statement and credit} Use normalized area measure on the unit disk $\mathbb D$, so $\|\sum a_kz^k\|_{A^2}^2=\sum_{k\ge0}|a_k|^2/(k+1)$. Let $C_{68}$ be the supremum of $c\in(0,1)$ such that \[ |f(z)|\le|u(z)|\quad(c<|z|<1),\quad f,u\in A^2(\mathbb D) \quad\Longrightarrow\quad \|f\|_{A^2}\le\|u\|_{A^2}. \] \begin{theorem} The radius $c=44667/100000$ is admissible. Thus $C_{68}\ge0.44667$. \end{theorem} David Niedbala Giraudin's PR~274 gives the Green-function flux input and the preceding $0.446$ bound, with a supplementary certificate at $0.44605$ \cite{flux}. The flux input is his, not a contribution of this note. The covering-strip coefficient approach continues Pinho's earlier construction; Wang supplies the coefficient-estimate precedent, and Wikstr\"om supplies moment integration and the atomic-plus-Lebesgue measure method \cite{wang,wikstrom,previous}. The added input is a nonconstant angular deficit converted into a low-coefficient correction. The supplied review supports this distinction: the present measure fails the old uncorrected first-moment condition. No optimality or exhaustive priority is asserted. \section{The coefficient correction} For $w=-\log c$, set \[ A=\pi^2/w,\quad q=e^{-A},\quad \Psi=\frac{(1+q)^2}{8q},\quad \sigma(t)=\sin^2\!\frac{\pi\log t}{2w},\quad \kappa=\frac3{25}, \] \begin{gather*} D(t)=\frac{\Psi}{2\sigma(t)},\quad H(t)=\sqrt{1+D(t)},\quad g(t)=\frac2{1+H(t)},\\ \beta(t)=1-g(t),\qquad \epsilon(t)=\frac7{100}\frac{D(t)}{H(t)(1+H(t))^2}. \end{gather*} At $t=c^2,1$, define $g=\epsilon=0$ by continuity. The certificate checks the four scalar hypotheses \begin{gather*} \operatorname{sech}^2(A/2)\le1/3,\qquad 16\sinh^2(A/4)\le2(1-8\kappa)\Psi,\\ \tanh^2(A/8)\ge(1+8\kappa)\pi^2/64,\qquad \operatorname{sech}^2(A/8)\le(1-4\kappa)/2. \end{gather*} Niedbala Giraudin's flux estimate, in curvature $-1$ hyperbolic distance, implies for a disk-valued annular map $\omega$ and two points on radius $r$ that \[ \cosh d(\omega(re^{i\theta}),\omega(re^{i\phi}))-1 \le\frac{2\Phi(A(\theta-\phi)/(2\pi))}{\sigma(r^2)}, \] with angles reduced modulo $2\pi$, where \[ \Phi(v)=\bigl(\operatorname{csch}^2v+ \operatorname{csch}^2(A-v)+2\operatorname{sech}^2(A/2)\bigr)^{-1}. \] The four hypotheses give the Fourier majorant \[ \frac{2\Phi(A\theta/(2\pi))}{\Psi} \le\frac{1-\cos\theta}{2}-\kappa(1-\cos2\theta). \] Its proof splits $0\le v\le A/2$ into a small arc, a midpoint neighborhood, and the intervening arc, using respectively $\Phi(v)\le\sinh^2v$, $\Phi(A/2+x)\le(\Psi/2)\operatorname{sech}^2x$, and the four displayed inequalities. The complete derivation, including flux normalization and the arc comparisons, is supplied in \texttt{GREEN\_FLUX\_PROOF.md} and \texttt{ANGULAR\_LEMMA.md}, Sections 1--2, in the accompanying analytic supplement \cite{certificate}. Distinct Green-function lifts are used. Choose the smallest enclosing hyperbolic disk of the image circle, center $\alpha$, radius $\rho$. Properness gives existence. First variation and separation give a probability measure on at most three contact directions balancing at zero. Let its domain angles be $\theta_i$, weights $\lambda_i$, and $m_j=\sum_i\lambda_i e^{ij\theta_i}$. The hyperbolic cosine rule, averaged over contacts, gives, with $h=\cosh\rho$ and $\zeta(\theta)=\cosh d(\omega(re^{i\theta}),\alpha)$, \[ H^2-\zeta(\theta)^2\ge D\max(S,T(\theta)), \] \[ S=|m_1|^2+2\kappa(1-|m_2|^2),\qquad T(\theta)=\Re(e^{i\theta}\overline m_1) +2\kappa\bigl(1-\Re(e^{2i\theta}\overline m_2)\bigr). \] Indeed the two contact averages give $h^2\le H^2-DS$ and $\zeta^2\le h\zeta\le1+2D\sum_i\lambda_i L(\theta-\theta_i)$, where $L=2\Phi/\Psi$. The Fourier majorant gives the other deficit $DT$. Put $a=|m_1|$, $b=|m_2|$. After an angular shift, \[ \max(S,T(\theta))\ge h_{ab}(\theta):= \max\left(a^2+\frac6{25}(1-b^2),\ \frac6{25}(1-b)+\frac{12}{25}b\sin^2\theta-a\right). \] For $a\ge27/100$ or $b\le21/25$, $S>7/100$ already. On the remaining rectangle, the scalar certificate proves $\exp(\int\log h_{ab}\,d\theta/(2\pi))\ge7/100$. The positive, $\pi$-periodic weight has an even analytic outer factor $C$, with $|C|^2=h_{ab}$ and $|C(0)|^2\ge7/100$. Hence $C_1=0$. Parseval applied to $CU(rz)$ gives \[ \int\max(S,T(\theta))|U(re^{i\theta})|^2\frac{d\theta}{2\pi} \ge\frac7{100}(|U_0|^2+r^2|U_1|^2). \] For $a=0$, the explicit factor of $(252/625)\sin^2\theta$ handles the logarithmic zeros. The small-$a$ proof and boundary limiting argument are given in \path{SPECTRAL_LEMMA.md} and \path{CONTACT_SPECTRAL_LEMMA.md}; both are included in full in the source archive. Only their scalar inequalities are delegated to computation. Now write $f=\sum a_kz^k$, $u=\sum b_kz^k$, $x_k=(|a_k|^2-|b_k|^2)_+$, $y_k=(|b_k|^2-|a_k|^2)_+$, $X(t)=\sum x_kt^k$, $Y(t)=\sum y_kt^k$, including $k=0$. The quotient $\omega=f/u$ is holomorphic on the annulus by removal of common zeros. Constant quotients give the norm comparison immediately. Otherwise use $F=f-\alpha u$, $U=u-\overline\alpha f$, analytic on the whole disk even if the quotient has inner poles. On the circle, $|F|^2=(1-v)|U|^2$, where $v=2/(1+\zeta)$, and \[ v-g\ge\frac{D\max(S,T)}{H(1+H)^2},\qquad |U_k|^2-|F_k|^2=(1-|\alpha|^2)(|b_k|^2-|a_k|^2). \] Moreover $|U_k|^2\ge(1-|\alpha|^2)y_k$. The preceding Parseval estimate therefore yields \begin{equation}\label{corrected} X(t)\le\beta(t)Y(t)-\epsilon(t)(y_0+y_1t). \end{equation} The inner endpoint follows by continuity. The checked inequality $\Psi>6$ implies $H>2$, $g<2/3$, and $0\le\beta-\epsilon\le1$. \section{Moments, tails, and the finite packet} The exact certificate specifies a probability measure \[ \mu=\sum_{j=1}^m w_j\delta_{t_j}+\mathbf1_{[R,1]}\,dt, \quad c^2=t_1<\cdots0,\quad\sum_jw_j=R. \] It suffices that \begin{gather*} \int t^k\,d\mu\ge\frac1{k+1}\quad(k\ge0),\qquad \int\beta(t)t^k\,d\mu\le\frac1{k+1}\quad(k\ge2),\\ \int(\beta-\epsilon)t\,d\mu\le\frac12. \end{gather*} The index-zero upper condition is automatic. Integrating \eqref{corrected} and applying Tonelli to its nonnegative coefficient weights gives $\sum x_k/(k+1)\le\int X\,d\mu\le\sum y_k/(k+1)$. Bergman Parseval proves the theorem. For explicit finite conditions, put $b_j(1)=\beta(t_j)-\epsilon(t_j)$, $b_j(k)=\beta(t_j)$ for $k\ge2$. The checked residuals are \[ \sum_jw_jt_j^k-\frac{R^{k+1}}{k+1}\ge0,\qquad \frac{R^{k+1}}{k+1}+\int_R^1 g(t)t^k\,dt -\sum_jw_jb_j(k)t_j^k\ge0. \] Discarding the continuous-tail $\epsilon$ term at $k=1$ weakens the sufficient condition. On each rational tail cell $[a,b]$, the proof derives $g(t)\ge\ell(1-t)$ using $-\log t\ge(1-t)(1+(1-b)/2)$ and $\sin x\ge x(1-U^2/6)$ for $0\le x\le U<\sqrt6$. The resulting polynomial integrals are exact. The first infinite tail follows from $w_m(K_1+1)\ge R$. For the second, a retained $R\le a\sum_jw_jt_j^{K_2}. \] Dividing by $a^k$, each atomic ratio $(t_j/a)^k$ is nonincreasing; the inequality persists for all $k\ge K_2$ and proves the second tail. The stored cutoffs are $K_1=6000$, $K_2=1800$. The precise packet is \texttt{contact-spectral} in \cite{certificate}: \texttt{certificate-contact-044667.json}, SHA-256 \begin{center}\scriptsize \texttt{aad4a43c5a7a72770dfced63b27466b66b4166dff5fad9ba1d86d8f0d184c83c}, \end{center} and \texttt{contact-cover.json}. The latter is a rational binary rectangle tree. Reconstruction checks coverage; each leaf uses a lower angular Riemann sum for $\log h_{ab}$. A separate analytic small-parameter estimate covers $0