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on Landau's constant ($C_{57b}$)

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 $L_f$ denote the radius of the largest disk contained in $f(\mathbb{D})$ . [BS2023-def-Lf]

The Landau constant is defined by

L\ :=\ \inf_{f\in\mathcal{F}} L_f.

[BS2023-def-L]

We define

C_{57b}\ :=\ L.

The best recorded bounds in the literature cited below are

\frac{1}{2}+10^{-335}\ <\ L\ \le\ \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\ \approx\ 0.5433.

[BS2023-bounds-L]

Moreover, Rademacher conjectured that the stated upper bound is the exact value of $L$ . [BS2023-rad-conj-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.