Research Problems in Function Theory — Problem 5.10

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 $0$. Let $S$ be the class of holomorphic injective functions $g:\mathbb D\to\mathbb C$ with $g(0)=0$ and $g'(0)=1$.Let $S_{\mathrm{star}}\subset S$ consist of those functions for which $g(\mathbb D)$ is starlike about $0$: $tw\in g(\mathbb D)$ whenever $w\in g(\mathbb D)$ and $0\le t\le1$. Put \[\ell(g)=\sup\{\rho>0:B(w,\rho)\subset g(\mathbb D)\text{ for some }w\in\mathbb C\},\qquad B_{\mathrm{sch}}=\inf_{g\in S}\ell(g),\quad B_{\mathrm{star}}=\inf_{g\in S_{\mathrm{star}}}\ell(g).\] The source denotes these two constants by $S$ and $S_1$, respectively.

Statement recorded in the supplied source.

Determine both $B_{\mathrm{sch}}$ and $B_{\mathrm{star}}$. [S2]

Short English statement

Find the exact universal image-disc radius for normalized univalent functions, and also for the subclass with starlike images.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305010 (AMR-022-5010). 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 5.10, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.