Research Problems in Function Theory — Problem 5.9

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 $f'(0)=1$.Let $\ell(f)=\sup\{\rho>0:B(w,\rho)\subset f(\mathbb D)\text{ for some }w\in\mathbb C\}$. Define Landau's constant $L=\inf_f\ell(f)$, with $f$ ranging over all normalized holomorphic functions on $\mathbb D$. Here the full image must contain the disc; no univalent inverse branch is required.

Statement recorded in the supplied source.

Determine the exact value of $L$. [S2]

Short English statement

Determine the sharp radius of a disc that must lie in the image of every normalized holomorphic disc function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305009 (AMR-022-5009). 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 5.9, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.