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OP-12071OpenAnalysis

Univalent Bloch constant (C57cC_{57c})

Univalent Bloch constant: determine C57cC_{57c} or improve its bounds; the repository records 0.5708858 (lower) and 1 (upper).

MSC 30C25 · Covering theorems in conformal mapping theoryMSC 30C45 · Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

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 BfB_f denote the radius of the largest univalent disk in f(D)f(\mathbb{D}). [BS2023-def-Bf]

The univalent Bloch constant is

Bu := inf⁡{Bf:f∈F, f is univalent in D}.B_u\ :=\ \inf\{B_f : f\in\mathcal{F},\ f\ \text{is univalent in }\mathbb{D}\}.

[BS2023-def-Bu]

We define

C57c := Bu.C_{57c}\ :=\ B_u.

The best rigorously stated lower bound in the cited survey literature is

Bu > 0.5708858.B_u\ >\ 0.5708858.

[BS2023-latest-Bu-lb]

Also, from B≤Bl≤L≤BuB\le B_l\le L\le B_u and L>12+10−335L>\frac{1}{2}+10^{-335}, one gets the weaker bound Bu>12+10−335B_u>\frac{1}{2}+10^{-335}. [BS2023-relations] [BS2023-bounds-L]

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Background

  • Relations to nearby constants. The standard chain is
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-Bf] 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}),"

  • [BS2023-def-Bu] loc: BS2023 PDF p.1270, §1 "Introduction" quote: "Bu:=inf⁡{Bf:f∈F,f is univalent in D}B_u := \inf\{B_f : f \in \mathcal{F}, f \text{ is univalent in } \mathbb{D}\},"

  • [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-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."

  • [BS2023-latest-Bu-lb] loc: BS2023 PDF p.1270, §1 "Introduction" quote: "In 1995, Yanagihara (see [12]) proved that Bl>1/2+10−335B_l > 1/2 + 10^{-335}. In 2009, Skinner (see [11]) proved that Bu>0.5708858B_u > 0.5708858. These bounds are latest bounds and best known so far."

  • [Skin2009] Skinner, Brian. The univalent Bloch constant problem. Complex Variables and Elliptic Equations 54 (2009), no. 10, 951–955. DOI: 10.1080/17476930903197199. 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 0.5708858; best known upper bound 1. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
11 Trivial The identity function f(z)=zf(z)=z is univalent with f′(0)=1f'(0)=1 and has Bf=1B_f=1, hence Bu≤1B_u\le 1.

Known lower bounds

Bound Reference Comments
12+10−335\dfrac{1}{2}+10^{-335} [Yan1995] Follows from L≤BuL\le B_u together with Yanagihara's lower bound for LL as summarized in [BS2023]. [BS2023-relations] [BS2023-bounds-L]
0.57088580.5708858 [Skin2009] Best recorded lower bound (as summarized in [BS2023]). [BS2023-latest-Bu-lb]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Univalent Bloch constant ($C_{57c}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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