Bloch’s constant

Description of constant

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, $f’(0)=1$). [BS2023-def-F]

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 $\Delta$). [BS2023-def-Bf] [BS2023-def-univalent-disk]

The Bloch constant is then defined by the extremal value \(B_{\mathrm{Bloch}}\ :=\ \inf_{f\in\mathcal{F}} B_f.\) [BS2023-def-B]

We define \(C_{57a}\ :=\ B_{\mathrm{Bloch}}.\)

The exact value of $B_{\mathrm{Bloch}}$ is not proved; the best recorded bounds in the literature cited below are \(\frac{\sqrt{3}}{4}+2\times 10^{-4}\ <\ B_{\mathrm{Bloch}}\ \le\ \frac{1}{\sqrt{1+\sqrt{3}}}\,\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\ \approx\ 0.4719.\) [BS2023-bounds-B]

Moreover, the upper bound is due to Ahlfors–Grunsky (1937) and was conjectured by them to be sharp. [BS2023-AG-conj-B]

Known upper bounds

Known lower bounds

Additional comments and links

• Conjectural value. It is conjectured that $B_{\mathrm{Bloch}}$ equals the Ahlfors–Grunsky upper bound listed above. [BS2023-AG-conj-B]

• 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 \(B_{\mathrm{Bloch}}\ \le\ L_{\mathrm{Landau}}\ \le\ B_u.\) [BS2023-relations]

• Wikipedia page on Bloch’s theorem

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: Cambridge PDF p.1, §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: Cambridge PDF p.1, §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-univalent-disk] loc: Cambridge PDF p.1, §1 “Introduction” 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 $\Delta$.”
  • [BS2023-def-B] loc: Cambridge PDF p.1, §1 “Introduction” quote: “$B:=\inf{B_f:f\in\mathcal{F}}$.”
  • [BS2023-bounds-B] loc: Cambridge PDF p.1, §1 “Introduction” quote: “the best known upper and lower bounds for $B$ are $\frac{\sqrt{3}}{4}+2\times10^{-4}<B\le \frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$.”
  • [BS2023-AG-conj-B] loc: Cambridge PDF p.1, §1 “Introduction” 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.”
  • [BS2023-relations] loc: Cambridge PDF p.2, §1 “Introduction” quote: “The relation between Bloch constant, Landau constant, locally univalent Bloch constant, and univalent Bloch constant is $B\le B_l\le L\le B_u$.”

• [AG1937] Ahlfors, Lars V.; Grunsky, Helmut. Über die Blochsche Konstante. Mathematische Zeitschrift 42 (1937), 671–673. DOI: 10.1007/BF01160101. Google Scholar

• [CG1996] Chen, Huaihui; Gauthier, Paul M. On Bloch’s constant. Journal d’Analyse Mathématique 69 (1996), 275–291. DOI: 10.1007/BF02787110. Google Scholar

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.