Univalent Bloch constant
Description of constant
Let $\mathbb{D}={z\in\mathbb{C}:\lvert z\rvert<1}$ and let $\mathcal{F}$ be the class of holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ normalized by $f’(0)=1$. [BS2023-def-F]
For $f\in\mathcal{F}$, let $B_f$ denote the radius of the largest univalent disk in $f(\mathbb{D})$. [BS2023-def-Bf]
The univalent Bloch constant is \(B_u\ :=\ \inf\{B_f : f\in\mathcal{F},\ f\ \text{is univalent in }\mathbb{D}\}.\) [BS2023-def-Bu]
We define \(C_{57c}\ :=\ B_u.\)
The best rigorously stated lower bound in the cited survey literature is \(B_u\ >\ 0.5708858.\) [BS2023-latest-Bu-lb]
Also, from $B\le B_l\le L\le B_u$ and $L>\frac{1}{2}+10^{-335}$, one gets the weaker bound $B_u>\frac{1}{2}+10^{-335}$. [BS2023-relations] [BS2023-bounds-L]
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial | The identity function $f(z)=z$ is univalent with $f’(0)=1$ and has $B_f=1$, hence $B_u\le 1$. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\dfrac{1}{2}+10^{-335}$ | [Yan1995] | Follows from $L\le B_u$ together with Yanagihara’s lower bound for $L$ as summarized in [BS2023]. [BS2023-relations] [BS2023-bounds-L] |
| $0.5708858$ | [Skin2009] | Best recorded lower bound (as summarized in [BS2023]). [BS2023-latest-Bu-lb] |
Additional comments and links
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Relations to nearby constants. The standard chain is \(B\ \le\ B_l\ \le\ L\ \le\ B_u.\) [BS2023-relations]
References
- [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 $\mathcal{F}$ be the set of all holomorphic functions from $\mathbb{D}$ to the complex plane $\mathbb{C}$ with $f’(0)=1$.”
- [BS2023-def-Bf] loc: BS2023 PDF p.1269, §1 “Introduction” quote: “Given a function $f \in \mathcal{F}$, let $B_f$ be the radius of the largest univalent disk in $f(\mathbb{D})$,”
- [BS2023-def-Bu] loc: BS2023 PDF p.1270, §1 “Introduction” quote: “$B_u := \inf{B_f : f \in \mathcal{F}, f \text{ is univalent in } \mathbb{D}}$,”
- [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 $\frac{1}{2}+10^{-335}<L\le \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$.”
- [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 $B\le B_l\le L\le B_u$.”
- [BS2023-latest-Bu-lb] loc: BS2023 PDF p.1270, §1 “Introduction” quote: “In 1995, Yanagihara (see [12]) proved that $B_l > 1/2 + 10^{-335}$. In 2009, Skinner (see [11]) proved that $B_u > 0.5708858$. These bounds are latest bounds and best known so far.”
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[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
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.