Submit progress

on Univalent Bloch constant ($C_{57c}$)

The statement

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]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

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