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Level A · Machine-checkable Hard Analysis P-bloch

Bloch and Landau constants

Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-bloch,
  title        = {Bloch and Landau constants},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/bloch}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

blochConstant_exact_value. Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.

landauConstant_exact_value. In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value of the Landau constant.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Bloch (2 statements).

theorem blochConstant_exact_value :
    blochConstant = Real.Gamma (1 / 3) * Real.Gamma (11 / 12) /
    (Real.Gamma (1 / 4) * Real.sqrt (1 + Real.sqrt 3))
theorem landauConstant_exact_value :
    landauConstant = Real.Gamma (1 / 3) * Real.Gamma (5 / 6) / Real.Gamma (1 / 6)

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • Wikipedia)
  • [CP96] Chen, H., Gauthier, P. M. "On Bloch's constant." Journal d'Analyse Mathématique 69 (1996), 275–291.
  • [AG37] Ahlfors, L. V., Grunsky, H. "Über die Blochsche Konstante." Mathematische Zeitschrift 42 (1937), 671–673.
  • [Ya95] Yanagihara, H. "On the locally univalent Bloch constant." Journal d'Analyse Mathématique 65 (1995), 1–17.
  • [Ra43] Rademacher, H. "On the Bloch-Landau Constant."" American Journal of Mathematics 65 (1943), 387–390.
  • OptimizationConstants
  • [Skin2009] Skinner, Brian. The univalent Bloch constant problem. Complex Variables and Elliptic Equations 54 (2009), no. 10, 951–955.
  • MathWorld
  • Bhowmik–Sen
  • [Re2008] Rettinger, R. "Bloch's constant is computable." Journal of Universal Computer Science 14 (2008), 896–907. In particular, a schlicht disk is the image of a subdomain under a biholomorphic restriction.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.