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A contact and coefficient correction for Korenblum's constant

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

Contributed by Ricardo Pinho

Version 3 / Oct 31, 2026 / CC BY 4.0

Abstract

We give an unrefereed computer-assisted proof that Korenblum's constant in unweighted Bergman space satisfies C68≥0.44685C_{68}\ge 0.44685. The new input retains the signed first contact harmonic in an annular coefficient estimate. An outer factor and square completion yield a correction of 19/20019/200 to the linear coefficient, replacing the 7/1007/100 correction of version 2. A rational measure verifies the resulting moment conditions. We include the analytic argument and describe the finite interval certificates and both infinite tails. The Green/flux input is V. Patankar's (PR 274); the coefficient and moment methods build on Wang, Pinho's earlier construction, and Wikström. There has been no independent human review or formalization.

Provenance statement

Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared certificates, checking software, and publication materials. No line-by-line human verification is claimed. AI is not an author.

References

  1. V. Patankar, A flux lemma and a half-diameter enclosing lemma for Korenblum's constant: C68≥0.446C_{68}\ge 0.446, proof, exact-rational certificates and two checkers (2026), https://github.com/vidigoat/optimization-records/tree/1a76f9dd053d51cce531b57b2ad5a68e3205ca75/tao/certificates/c68; Tao-table PR 274 (open, unrefereed), https://github.com/teorth/optimizationproblems/pull/274.link
  2. C. Wang, A lower bound on Korenblum's constant, Proc. Amer. Math. Soc. 153 (2025), 1209--1214, doi:10.1090/proc/17112 (https://doi.org/10.1090/proc/17112); Domination in the Bergman space and Korenblum's constant, Integral Equations Operator Theory 61 (2008), 423--432, https://doi.org/10.1007/s00020-008-1587-4.DOI
  3. A. Schuster, The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus, Proc. Amer. Math. Soc. 134 (2006), 3525--3530, doi:10.1090/S0002-9939-06-08378-X (https://doi.org/10.1090/S0002-9939-06-08378-X).DOI
  4. F. Wikström, Moment duality and an improved lower bound for Korenblum's constant, arXiv:2607.17748 (https://arxiv.org/abs/2607.17748) (2026), especially Lemma 3.1.arXiv
  5. R. Pinho, A covering-strip and moment certificate for Korenblum's constant, unrefereed preprint, 2026; derivation and certificates at the pinned commit (https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c68).link
  6. R. Pinho, A contact and coefficient correction for Korenblum's constant, hexagon:2610.00696v2 (https://hexagonmath.org/2610.00696?version=2), unrefereed preprint, 2026; earlier contact-spectral packet at the pinned commit (https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c68/contact-spectral).hexagon
  7. T. Ransford, Potential Theory in the Complex Plane , London Mathematical Society Student Texts 28, Cambridge University Press, 1995.
  8. T. Tao et al., Optimization problems, constant 68, https://teorth.github.io/optimizationproblems/constants/68a.html, checked 2 November 2026.link
  9. R. Pinho, C68: lower bound 0.44685 by a signed contact-harmonic lemma (unrefereed), PR 299, https://github.com/teorth/optimizationproblems/pull/299.link
  10. R. Pinho, Korenblum C68 signed-contact certificate, model signed_contact_v4 , code, proofs, certificates and receipts (https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c68/signed-contact), commit 848f899c973666f64b73fb6f37760849a600285f .link

Version history

  1. v1Submitted by Ricardo PinhoInitial depositSupersededOct 30, 2026
  2. v2Submitted by Ricardo PinhoSubmitted as a new versionSupersededOct 31, 2026
  3. v3Submitted by Ricardo PinhoSubmitted as a new versionCurrentNov 03, 2026