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OP-12069OpenAnalysis

Bloch’s constant (C57aC_{57a})

Bloch’s constant: determine C57aC_{57a} or improve its bounds; the repository records 34+2×10−4\frac{\sqrt{3}}{4}+2\times 10^{-4} (lower) and 11+3 Γ(1/3)Γ(11/12)Γ(1/4)≈0.4719\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719 (upper).

MSC 30C25 · Covering theorems in conformal mapping theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

Let D={z∈C:∣z∣<1}\mathbb{D}=\{z\in\mathbb{C}:\lvert z\rvert<1\}. Following standard notation, let F\mathcal{F} be the class of holomorphic functions f:D→Cf:\mathbb{D}\to\mathbb{C} normalized by ∣f′(0)∣=1\lvert f'(0)\rvert=1 (equivalently, after rotation, f′(0)=1f'(0)=1). [BS2023-def-F]

For f∈Ff\in\mathcal{F}, let BfB_f denote the radius of the largest univalent disk contained in f(D)f(\mathbb{D}) (i.e., a disk Δ⊂f(D)\Delta\subset f(\mathbb{D}) such that some domain Ω⊂D\Omega\subset\mathbb{D} is mapped univalently by ff onto Δ\Delta). [BS2023-def-Bf] [BS2023-def-univalent-disk]

The Bloch constant is then defined by the extremal value

BBloch := inf⁡f∈FBf.B_{\mathrm{Bloch}}\ :=\ \inf_{f\in\mathcal{F}} B_f.

[BS2023-def-B]

We define

C57a := BBloch.C_{57a}\ :=\ B_{\mathrm{Bloch}}.

The exact value of BBlochB_{\mathrm{Bloch}} is not proved; the best recorded bounds in the literature cited below are

34+2×10−4 < BBloch ≤ 11+3 Γ(1/3)Γ(11/12)Γ(1/4) ≈ 0.4719.\frac{\sqrt{3}}{4}+2\times 10^{-4}\ <\ B_{\mathrm{Bloch}}\ \le\ \frac{1}{\sqrt{1+\sqrt{3}}}\,\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\ \approx\ 0.4719.

[BS2023-bounds-B]

Moreover, the upper bound is due to Ahlfors–Grunsky (1937) and was conjectured by them to be sharp. [BS2023-AG-conj-B]

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Background

  • Conjectural value. It is conjectured that BBlochB_{\mathrm{Bloch}} equals the Ahlfors–Grunsky upper bound listed above. [BS2023-AG-conj-B]

  • Relation to Landau-type constants. If LLandauL_{\mathrm{Landau}} is Landau’s constant (entry 57b) and BuB_u is the univalent Bloch constant (entry 57c), then

BBloch ≤ LLandau ≤ Bu.B_{\mathrm{Bloch}}\ \le\ L_{\mathrm{Landau}}\ \le\ B_u.

[BS2023-relations]

References (from the repository page)

  • [BS2023] Bhowmik, Bappaditya; Sen, Sambhunath. Improved Bloch and Landau constants for meromorphic functions. Canadian Mathematical Bulletin 66 (2023), 1269–1273. DOI: 10.4153/S0008439523000346. PDF: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FD465D1F2CEF7E8C62AFF16C3E89B7B4/S0008439523000346a.pdf/improved_bloch_and_landau_constants_for_meromorphic_functions.pdf. Google Scholar

  • [BS2023-def-F] loc: Cambridge PDF p.1, §1 “Introduction” quote: “let F\mathcal{F} be the set of all holomorphic functions from D\mathbb{D} to the complex plane C\mathbb{C} with f′(0)=1f'(0)=1.”

  • [BS2023-def-Bf] loc: Cambridge PDF p.1, §1 “Introduction” quote: “Given a function f∈Ff\in\mathcal{F}, let BfB_f be the radius of the largest univalent disk in f(D)f(\mathbb{D}),”

  • [BS2023-def-univalent-disk] loc: Cambridge PDF p.1, §1 “Introduction” quote: “by a univalent disk Δ\Delta in f(D)f(\mathbb{D}), we mean that there exists a domain Ω\Omega in D\mathbb{D} such that ff maps Ω\Omega univalently onto Δ\Delta.”

  • [BS2023-def-B] loc: Cambridge PDF p.1, §1 “Introduction” quote: “B:=inf⁡{Bf:f∈F}B:=\inf\{B_f:f\in\mathcal{F}\}.”

  • [BS2023-bounds-B] loc: Cambridge PDF p.1, §1 “Introduction” quote: “the best known upper and lower bounds for BB are 34+2×10−4<B≤11+3Γ(1/3)Γ(11/12)Γ(1/4)≈0.4719\frac{\sqrt{3}}{4}+2\times10^{-4}<B\le \frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719.”

  • [BS2023-AG-conj-B] loc: Cambridge PDF p.1, §1 “Introduction” quote: “The upper bound for the Bloch constant BB was obtained by Ahlfors and Grunsky; also, they conjectured that this upper bound is the precise value.”

  • [BS2023-relations] loc: Cambridge PDF p.2, §1 “Introduction” quote: “The relation between Bloch constant, Landau constant, locally univalent Bloch constant, and univalent Bloch constant is B≤Bl≤L≤BuB\le B_l\le L\le B_u.”

  • [AG1937] Ahlfors, Lars V.; Grunsky, Helmut. Über die Blochsche Konstante. Mathematische Zeitschrift 42 (1937), 671–673. DOI: 10.1007/BF01160101. Google Scholar

  • [CG1996] Chen, Huaihui; Gauthier, Paul M. On Bloch’s constant. Journal d’Analyse Mathématique 69 (1996), 275–291. DOI: 10.1007/BF02787110. Google Scholar

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 34+2×10−4\frac{\sqrt{3}}{4}+2\times 10^{-4}; best known upper bound 11+3 Γ(1/3)Γ(11/12)Γ(1/4)≈0.4719\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
11+3 Γ(1/3)Γ(11/12)Γ(1/4)≈0.4719\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719 [AG1937] Ahlfors–Grunsky bound; conjectured sharp. [BS2023-AG-conj-B] [BS2023-bounds-B]

Known lower bounds

Bound Reference Comments
34+2×10−4\dfrac{\sqrt{3}}{4}+2\times 10^{-4} [CG1996] Best recorded lower bound (as quoted in the survey literature). [BS2023-bounds-B]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Bloch’s constant ($C_{57a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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