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OP-12070OpenAnalysis

Landau's constant (C57bC_{57b})

Landau's constant: determine C57bC_{57b} or improve its bounds; the repository records 12+10−335\frac{1}{2}+10^{-335} (lower) and Γ(1/3)Γ(5/6)Γ(1/6)≈0.5433\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433 (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 ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

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

For f∈Ff\in\mathcal{F}, let LfL_f denote the radius of the largest disk contained in f(D)f(\mathbb{D}). [BS2023-def-Lf]

The Landau constant is defined by

L := inf⁡f∈FLf.L\ :=\ \inf_{f\in\mathcal{F}} L_f.

[BS2023-def-L]

We define

C57b := L.C_{57b}\ :=\ L.

The best recorded bounds in the literature cited below are

12+10−335 < L ≤ Γ(1/3)Γ(5/6)Γ(1/6) ≈ 0.5433.\frac{1}{2}+10^{-335}\ <\ L\ \le\ \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\ \approx\ 0.5433.

[BS2023-bounds-L]

Moreover, Rademacher conjectured that the stated upper bound is the exact value of LL. [BS2023-rad-conj-L]

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Background

  • Relations to nearby constants. If BB is the Bloch constant (entry 57a), BlB_l the locally univalent Bloch constant, and BuB_u the univalent Bloch constant (entry 57c), then
B ≤ Bl ≤ L ≤ Bu.B\ \le\ B_l\ \le\ L\ \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: BS2023 PDF p.1269, §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-Lf] loc: BS2023 PDF p.1269, §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}), and let LfL_f be the radius of the largest disk in f(D)f(\mathbb{D})."

  • [BS2023-def-L] loc: BS2023 PDF p.1269, §1 "Introduction" quote: "L:=inf⁡{Lf:f∈F}L := \inf\{L_f : f \in \mathcal{F}\}."

  • [BS2023-bounds-L] loc: BS2023 PDF p.1270, §1 "Introduction" quote: "Rademacher (compare [10]) and Yanagihara (in 1995, see [12]) proved that the upper and the lower bounds for the Landau constant are 12+10−335<L≤Γ(1/3)Γ(5/6)Γ(1/6)≈0.5433\frac{1}{2}+10^{-335}<L\le \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433."

  • [BS2023-rad-conj-L] loc: BS2023 PDF p.1270, §1 "Introduction" quote: "Rademacher (compare [10]) also conjectured that this upper bound is the precise value of the Landau constant."

  • [BS2023-relations] loc: BS2023 PDF p.1270, §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."

  • [Rad1943] Rademacher, Hans. On the Bloch-Landau constant. American Journal of Mathematics 65 (1943), no. 3, 387–390. DOI: 10.2307/2371963. Google Scholar

  • [Yan1995] Yanagihara, H. On the locally univalent Bloch constant. Journal d'Analyse Mathématique 65 (1995), 1–17. DOI: 10.1007/BF02788763. Google Scholar

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 12+10−335\frac{1}{2}+10^{-335}; best known upper bound Γ(1/3)Γ(5/6)Γ(1/6)≈0.5433\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
Γ(1/3)Γ(5/6)Γ(1/6)≈0.5433\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433 [Rad1943] Upper bound attributed to Rademacher (as summarized in [BS2023]). [BS2023-bounds-L]

Known lower bounds

Bound Reference Comments
12+10−335\dfrac{1}{2}+10^{-335} [Yan1995] Lower bound attributed to Yanagihara (as summarized in [BS2023]). [BS2023-bounds-L]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Landau's constant ($C_{57b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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