Bloch and Landau constants

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

Current state (tracked automatically)

Best known bounds for constant 57c: 0.5708858 ≤ C ≤ 0.65628589. Upper bound ([vS2026], Disc of radius R=5.001117628756668 minus 24 slits z= w, w= R+s+i(+b\_1s^2+b\_2s^4), (t/R)≤ s≤ 0, with (,t,b\_1,b\_2) equal to (0,1,0,0), (/3,1.9999999974092333,), not yet independently verified; lower bound ([Skin2009], Best recorded lower bound (as summarized in [BS2023])).
The best independently verified upper bound is 0.6563937 ([COC2009], Disc of radius 4.0546358 minus six radial slits and six harmonically symmetric arcs).

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 6 solved	Formal Conjectures
Best lower bound with an explicit value (constant 57c)	0.5708858 [Skin2009] — Best recorded lower bound (as summarized in [BS2023]).	Tao's optimization constants
Best upper bound with an explicit value (constant 57c)	0.65628589* [vS2026] — Disc of radius R=5.001117628756668 minus 24 slits z= w, w= R+s+i(+b\_1s^2+b\_2s^4), (t/R)≤ s≤ 0, with (,t,b\_1,b\_2) equal to (0,1,0,0), (/3,1.9999999974092333,	Tao's optimization constants
Best upper bound with an explicit value, independently verified (constant 57c)	0.6563937 [COC2009] — Disc of radius 4.0546358 minus six radial slits and six harmonically symmetric arcs;	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12), Tao's optimization constants (checked 2026-11-12). How it works.

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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.