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on Bloch’s constant ($C_{57a}$)

The statement

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]

What gets accepted

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Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.