Research Problems in Function Theory — Problem 5.5

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 $0$. For holomorphic $f(z)=\sum_{n=0}^\infty a_nz^n$ on $\mathbb D$, put \[d_f(R)=\sup\{\rho>0:B(w,\rho)\subset f(\mathbb D)\text{ for some }|w|=R\}\quad(R>0),\] where $B(w,\rho)=\{v:|v-w|<\rho\}$ and the supremum of the empty set is $0$. Thus this measures the largest full image disc whose center has modulus $R$. The assertion $a_n=O(1)$ means $\sup_{n\ge1}|a_n|<\infty$ for the particular function.

Statement recorded in the supplied source.

Describe conditions on an image domain $D\subset\mathbb C$ ensuring that every holomorphic $f:\mathbb D\to D$ has $a_n=O(1)$. In particular, does boundedness of the coefficients still follow when $d_f(R)$ is allowed to tend to infinity sufficiently slowly? Formulate the allowable growth rates: determine whether there is a function $h:[1,\infty)\to(0,\infty)$, with $h(R)\to\infty$, for which $d_f(R)\le h(R)$ for all sufficiently large $R$ forces $a_n=O(1)$ for every such $f$. The bound for the coefficients may depend on $f$. [S2]

Short English statement

Which restrictions on the image of a disc function force bounded Taylor coefficients, and can the size of full image discs grow slowly without losing that conclusion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305005 (AMR-022-5005). 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 5.5, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.