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A covering-strip and moment certificate for Korenblum's constant

Primarily AI-generated textHuman understanding: some partsmath.CV — Complex Variables

Contributed by Ricardo Pinho ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We give an explicit certificate for c_2>=54/125=0.432, where c_2 is Korenblum's constant for the Bergman space. This is an unrefereed preprint. The construction combines an elementary two-zero Schwarz estimate on the covering strip of an annulus, the classical hyperbolic Jung inequality, and Wikström's moment-duality criterion. A rational twelve-atom measure with a Lebesgue tail satisfies the required inequalities for every moment index. Two independently implemented exact checkers, one using Arb and one using only integer interval arithmetic, verify the quantitative hypotheses. The best public certified lower bound located in the accompanying priority audit is Wikström's 0.4263. Neither optimality nor a new version of Jung's theorem is claimed.

Provenance statement

Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared the certificate, checking software, and publication materials. This does not claim a line-by-line human verification of the proof or software. The quantitative hypotheses are verified by two independently implemented exact checkers (Arb ball arithmetic; pure integer interval arithmetic), with 18 mutation/control runs rejected as intended. Repository: https://github.com/rpinho/korenblum-c68-certificate (commit 9d48de64987911f5a07a397323345be8b97770a8); the checkers and certificates are also included in anc/.

Tools used

OpenAI
CodexVersion GPT-6.1 Sol
FLINT
Arb ball arithmetic (python-flint)Version python-flint 0.9.0

References

  1. B. Korenblum, A maximum principle for the Bergman space , Publ. Mat. 35 (1991), no. 2, 479--486.
  2. W. K. Hayman, On a conjecture of Korenblum , Analysis (Munich) 19 (1999), no. 2, 195--205.
  3. F. Wikström, Moment duality and an improved lower bound for Korenblum's constant , arXiv:2607.17748 (2026), https://arxiv.org/abs/2607.17748.arXiv
  4. F. Wikström, Rigorous certificates for a lower bound on Korenblum's constant , Zenodo, doi:10.5281/zenodo.21446369.DOI
  5. 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
  6. C. Wang, A lower bound on Korenblum's constant , Proc. Amer. Math. Soc. 153 (2025), 1209--1214, doi:10.1090/proc/17112.DOI
  7. A. Schuster, The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus , Proc. Amer. Math. Soc. 134 (2006), 3525--3530; arXiv:math/0407416.arXiv
  8. S. Baguley, Y. Maus, J. Ruff and G. Skretas, Hyperbolic random graphs: clique number and degeneracy with implications for colouring , STACS 2025, LIPIcs 327 , article 13, Theorem 17 (hyperbolic Jung, citing Dekster), doi:10.4230/LIPIcs.STACS.2025.13.DOI

Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 19, 2026