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

Primarily AI-generated textHuman understanding: no partsmath.CV — Complex Variablesmath.FA — Functional Analysis

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We show that Korenblum's constant of the Bergman space satisfies c2≥0.446c_2\ge 0.446. The best published lower bound was c2≥0.4263c_2\ge 0.4263 (Wikström, 2026). The new ingredient is a closed form, in Jacobi theta functions, of the Möbius pseudodistance of the annulus {c<∣z∣<1}\{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. Wikström's dual moment criterion is then verified at c=223/500c=223/500 for an explicit measure with fourteen rational atoms, in rigorous arithmetic, by a self-contained verifier. The French version of this note is dated 21 October 2026.

Provenance statement

Mathematical content, LaTeX source and verifier produced with Claude (Anthropic) under the author's direction. French original dated 21 October 2026 (OpenTimestamps); English version 1 November 2026.

Tools used

Anthropic
Claude Opus

References

  1. NIST Digital Library of Mathematical Functions, Chapters 20 and 22, https://dlmf.nist.gov.link
  2. W. K. Hayman, On a conjecture of Korenblum , Analysis (Munich) 19 (1999), 195--205.
  3. B. Korenblum, A maximum principle for the Bergman space , Publ.\ Mat.\ 35 (1991), 479--486.
  4. U. Lang, V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature , Geom.\ Funct.\ Anal.\ 7 (1997), 535--560.
  5. R. Pinho, Korenblum's constant: a certificate for C68≥0.432C_{68}\ge0.432 , hexagon:2610.00399v1, https://github.com/rpinho/korenblum-c68-certificate.hexagon
  6. R. Pinho, A contact and coefficient correction for Korenblum's constant , hexagon:2610.00696v1 (31 October 2026), https://hexagonmath.org/2610.00696v1.hexagon
  7. vidigoat , pull request #274 to teorth/optimizationproblems , https://github.com/teorth/optimizationproblems/pull/274.link
  8. A. Schuster, The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus , Proc.\ Amer.\ Math.\ Soc.\ 134 (2006), no. 12, 3525--3530, doi:10.1090/S0002-9939-06-08378-X.DOI
  9. F. A. Valentine, Contractions in non-Euclidean spaces , Bull.\ Amer.\ Math.\ Soc.\ 50 (1944), 710--713.
  10. C. Wang, Domination in the Bergman space and Korenblum's constant , Integral Equations Operator Theory 61 (2008), 423--432, doi:10.1007/s00020-008-1587-4.DOI
Show all 12 referencesShow fewer references
  1. C. Wang, A lower bound on Korenblum's constant , Proc.\ Amer.\ Math.\ Soc.\ 153 (2025), 1209--1214.
  2. F. Wikström, Moment duality and an improved lower bound for Korenblum's constant , arXiv:2607.17748 (2026).arXiv

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 01, 2026