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 $B_u\le 1$. |
| $0.658$ | [Rob1935] | Disc of radius about $2.956$ minus six radial slits ending at moduli $1$ and $2$. |
| $0.65647$ | [Good1945] | Radial slits added in bisecting generations. |
| $0.6564155$ | [BH1985] | Radial slits; third generation not bisecting. Numerical. |
| $0.6563937$ | [COC2009] | Disc of radius $4.0546358$ minus six radial slits and six harmonically symmetric arcs; numerical evaluation of an explicit theta-function formula, stated correct to 10 decimal places. |
| $0.656319277272^*$ | [OCP2010] | Disc of radius $5.1195152501$ minus six radial slits and further harmonically symmetric arcs; computed numerically, and the authors state that no rigorous error analysis was done. |
| $0.65628589^*$ | [vS2026] | Disc of radius $R=5.001117628756668$ minus 24 slits $z=\exp w$, $w=\log R+s+i(\theta+b_1s^2+b_2s^4)$, $\log(t/R)\le s\le 0$, with $(\theta,t,b_1,b_2)$ equal to $(0,1,0,0)$, $(\pi/3,1.9999999974092333,0,0)$, $(0.029930188727091613,2.789342409403121,0.9026921410641155,-0.3301459494660587)$, $(1.0157094972537686,3.473381248030611,-1.2656124700253997,0.9794269477754911)$, $(0.5757776442359243,4.342874184977691,0.03225521024585182,2.9360081049550737)$, and their images under rotation by $2\pi/3$ and complex conjugation. Computer-assisted proof: inradius $\le 1+10^{-9}$ (exact covering), conformal radius $\ge 1.523726208329$ (comparison function, Arb ball arithmetic). Floating-point value of the ratio $0.656285866$. |
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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[BH1985] Beller, E.; Hummel, J. A. On the univalent Bloch constant. Complex Variables Theory Appl. 4 (1985), no. 3, 243–252. DOI: 10.1080/17476938508814109.
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[COC2009] Carroll, T.; Ortega-Cerdà, J. The univalent Bloch–Landau constant, harmonic symmetry and conformal glueing. J. Math. Pures Appl. (9) 92 (2009), no. 4, 396–406. DOI: 10.1016/j.matpur.2009.05.008. arXiv:0806.2282.
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[Good1945] Goodman, R. E. On the Bloch–Landau constant for schlicht functions. Bull. Amer. Math. Soc. 51 (1945), no. 4, 234–239. DOI: 10.1090/S0002-9904-1945-08315-3.
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[OCP2010] Ortega-Cerdà, J.; Pridhnani, B. The Pólya–Tchebotaröv problem. In: Harmonic analysis and partial differential equations, Contemp. Math. 505, Amer. Math. Soc., 2010, 153–170. DOI: 10.1090/conm/505/09921. arXiv:0809.2483.
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[Rob1935] Robinson, R. M. The Bloch constant 𝔄 for a schlicht function. Bull. Amer. Math. Soc. 41 (1935), no. 8, 535–540. DOI: 10.1090/S0002-9904-1935-06138-5.
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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
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[Yan1995] Yanagihara, H. On the locally univalent Bloch constant. Journal d’Analyse Mathématique 65 (1995), 1–17. DOI: 10.1007/BF02788763. Google Scholar
- [vS2026] van Staden, W. An improved upper bound for the univalent Bloch–Landau constant. Preprints.org, 20 October 2026: preprints.org/manuscript/202610.1936/v1, DOI 10.20944/preprints202610.1936.v1. Code and certificates: github.com/WynandVStaden/bloch-landau-slit-discs.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.
The history of upper bounds and the bound of [vS2026] were added with the help of Claude (Anthropic). The bound of [vS2026] has not been peer reviewed.