Research Problems in Function Theory — Problem 5.8

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$. A normalized holomorphic function has $f(0)=0$ and $f'(0)=1$.For such a function, let \[b(f)=\sup\{\rho>0: f|_U:U\longrightarrow B(w,\rho)\text{ is biholomorphic for some open }U\subset\mathbb D\text{ and }w\in\mathbb C\}.\] A biholomorphic map is a holomorphic bijection with holomorphic inverse. Define Bloch's constant by $B=\inf_f b(f)$, the infimum over all normalized holomorphic functions on $\mathbb D$. Thus a disc counted in $b(f)$ must admit one univalent inverse branch. The original collection records the historical bounds $\sqrt3/4\le B<0.472$ and attributes its upper estimate to Ahlfors and Grunsky.

Statement recorded in the supplied source.

Determine the exact value of $B$, the sharp universal lower bound for the radii $b(f)$. [S2]

Short English statement

Determine the sharp universal radius of a disc covered by a single univalent branch of a normalized holomorphic function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305008 (AMR-022-5008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.8, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.