|
| 1 | +# Bloch’s constant |
| 2 | + |
| 3 | +## Description of constant |
| 4 | + |
| 5 | +Let $\mathbb{D}=\{z\in\mathbb{C}:\lvert z\rvert<1\}$. Following standard notation, let $\mathcal{F}$ be the class of holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ normalized by $\lvert f'(0)\rvert=1$ (equivalently, after rotation, $f'(0)=1$). <a href="#BS2023-def-F">[BS2023-def-F]</a> |
| 6 | + |
| 7 | +For $f\in\mathcal{F}$, let $B_f$ denote the radius of the largest **univalent disk** contained in $f(\mathbb{D})$ (i.e., a disk $\Delta\subset f(\mathbb{D})$ such that some domain $\Omega\subset\mathbb{D}$ is mapped univalently by $f$ onto $\Delta$). <a href="#BS2023-def-Bf">[BS2023-def-Bf]</a> <a href="#BS2023-def-univalent-disk">[BS2023-def-univalent-disk]</a> |
| 8 | + |
| 9 | +The **Bloch constant** is then defined by the extremal value |
| 10 | +$$ |
| 11 | +B_{\mathrm{Bloch}}\ :=\ \inf_{f\in\mathcal{F}} B_f. |
| 12 | +$$ |
| 13 | +<a href="#BS2023-def-B">[BS2023-def-B]</a> |
| 14 | + |
| 15 | +We define |
| 16 | +$$ |
| 17 | +C_{57a}\ :=\ B_{\mathrm{Bloch}}. |
| 18 | +$$ |
| 19 | + |
| 20 | +The exact value of $B_{\mathrm{Bloch}}$ is not proved; the best recorded bounds in the literature cited below are |
| 21 | +$$ |
| 22 | +\frac{\sqrt{3}}{4}+2\times 10^{-4}\ <\ B_{\mathrm{Bloch}}\ \le\ \frac{1}{\sqrt{1+\sqrt{3}}}\,\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\ \approx\ 0.4719. |
| 23 | +$$ |
| 24 | +<a href="#BS2023-bounds-B">[BS2023-bounds-B]</a> |
| 25 | + |
| 26 | +Moreover, the upper bound is due to Ahlfors–Grunsky (1937) and was conjectured by them to be sharp. <a href="#BS2023-AG-conj-B">[BS2023-AG-conj-B]</a> |
| 27 | + |
| 28 | +## Known upper bounds |
| 29 | + |
| 30 | +| Bound | Reference | Comments | |
| 31 | +| ----- | --------- | -------- | |
| 32 | +| $\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$ | <a href="#AG1937">[AG1937]</a> | Ahlfors–Grunsky bound; conjectured sharp. <a href="#BS2023-AG-conj-B">[BS2023-AG-conj-B]</a> <a href="#BS2023-bounds-B">[BS2023-bounds-B]</a> | |
| 33 | + |
| 34 | +## Known lower bounds |
| 35 | + |
| 36 | +| Bound | Reference | Comments | |
| 37 | +| ----- | --------- | -------- | |
| 38 | +| $\dfrac{\sqrt{3}}{4}+2\times 10^{-4}$ | <a href="#CG1996">[CG1996]</a> | Best recorded lower bound (as quoted in the survey literature). <a href="#BS2023-bounds-B">[BS2023-bounds-B]</a> | |
| 39 | + |
| 40 | +## Additional comments and links |
| 41 | + |
| 42 | +- **Conjectural value.** It is conjectured that $B_{\mathrm{Bloch}}$ equals the Ahlfors–Grunsky upper bound listed above. <a href="#BS2023-AG-conj-B">[BS2023-AG-conj-B]</a> |
| 43 | + |
| 44 | +- **Relation to Landau-type constants.** If $L_{\mathrm{Landau}}$ is Landau’s constant (entry 57b) and $B_u$ is the univalent Bloch constant (entry 57c), then |
| 45 | + $$ |
| 46 | + B_{\mathrm{Bloch}}\ \le\ L_{\mathrm{Landau}}\ \le\ B_u. |
| 47 | + $$ |
| 48 | + <a href="#BS2023-relations">[BS2023-relations]</a> |
| 49 | + |
| 50 | +- [Wikipedia page on Bloch’s theorem](https://en.wikipedia.org/wiki/Bloch%27s_theorem) |
| 51 | + |
| 52 | +## References |
| 53 | + |
| 54 | +- <a id="BS2023"></a>**[BS2023]** Bhowmik, Bappaditya; Sen, Sambhunath. *Improved Bloch and Landau constants for meromorphic functions.* Canadian Mathematical Bulletin **66** (2023), 1269–1273. DOI: [10.4153/S0008439523000346](https://doi.org/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](https://scholar.google.com/scholar?q=Improved+Bloch+and+Landau+constants+for+meromorphic+functions+Bhowmik+Sen+2023) |
| 55 | + - <a id="BS2023-def-F"></a>**[BS2023-def-F]** |
| 56 | + **loc:** Cambridge PDF p.1, §1 “Introduction” |
| 57 | + **quote:** “let $\mathcal{F}$ be the set of all holomorphic functions from $\mathbb{D}$ to the complex plane $\mathbb{C}$ with $f'(0)=1$.” |
| 58 | + - <a id="BS2023-def-Bf"></a>**[BS2023-def-Bf]** |
| 59 | + **loc:** Cambridge PDF p.1, §1 “Introduction” |
| 60 | + **quote:** “Given a function $f\in\mathcal{F}$, let $B_f$ be the radius of the largest univalent disk in $f(\mathbb{D})$,” |
| 61 | + - <a id="BS2023-def-univalent-disk"></a>**[BS2023-def-univalent-disk]** |
| 62 | + **loc:** Cambridge PDF p.1, §1 “Introduction” |
| 63 | + **quote:** “by a univalent disk $\Delta$ in $f(\mathbb{D})$, we mean that there exists a domain $\Omega$ in $\mathbb{D}$ such that $f$ maps $\Omega$ univalently onto $\Delta$.” |
| 64 | + - <a id="BS2023-def-B"></a>**[BS2023-def-B]** |
| 65 | + **loc:** Cambridge PDF p.1, §1 “Introduction” |
| 66 | + **quote:** “$B:=\inf\{B_f:f\in\mathcal{F}\}$.” |
| 67 | + - <a id="BS2023-bounds-B"></a>**[BS2023-bounds-B]** |
| 68 | + **loc:** Cambridge PDF p.1, §1 “Introduction” |
| 69 | + **quote:** “the best known upper and lower bounds for $B$ are $\frac{\sqrt{3}}{4}+2\times10^{-4}<B\le \frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$.” |
| 70 | + - <a id="BS2023-AG-conj-B"></a>**[BS2023-AG-conj-B]** |
| 71 | + **loc:** Cambridge PDF p.1, §1 “Introduction” |
| 72 | + **quote:** “The upper bound for the Bloch constant $B$ was obtained by Ahlfors and Grunsky; also, they conjectured that this upper bound is the precise value.” |
| 73 | + - <a id="BS2023-relations"></a>**[BS2023-relations]** |
| 74 | + **loc:** Cambridge PDF p.2, §1 “Introduction” |
| 75 | + **quote:** “The relation between Bloch constant, Landau constant, locally univalent Bloch constant, and univalent Bloch constant is $B\le B_l\le L\le B_u$.” |
| 76 | + |
| 77 | +- <a id="AG1937"></a>**[AG1937]** Ahlfors, Lars V.; Grunsky, Helmut. *Über die Blochsche Konstante.* Mathematische Zeitschrift **42** (1937), 671–673. DOI: [10.1007/BF01160101](https://doi.org/10.1007/BF01160101). [Google Scholar](https://scholar.google.com/scholar?q=Ahlfors+Grunsky+%C3%9Cber+die+Blochsche+Konstante+Math.+Z.+42+1937+671-673) |
| 78 | + |
| 79 | +- <a id="CG1996"></a>**[CG1996]** Chen, Huaihui; Gauthier, Paul M. *On Bloch’s constant.* Journal d’Analyse Mathématique **69** (1996), 275–291. DOI: [10.1007/BF02787110](https://doi.org/10.1007/BF02787110). [Google Scholar](https://scholar.google.com/scholar?q=Chen+Gauthier+On+Bloch%27s+constant+J.+Analyse+Math.+69+1996+275-291) |
| 80 | + |
| 81 | +## Contribution notes |
| 82 | + |
| 83 | +Prepared with assistance from ChatGPT 5.2 Pro. |
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