Research Problems in Function Theory — Problem 5.7

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.

Find nontrivial conditions on an omitted set $E\subset\mathbb C$ such that every holomorphic $f:\mathbb D\to\mathbb C\setminus E$ has $a_n\to0$. In particular, for arbitrary holomorphic $f$ on $\mathbb D$, does \[d_f(R)\longrightarrow0\quad(R\to\infty)\qquad\Longrightarrow\qquad a_n\longrightarrow0\quad(n\to\infty)?\] The question concerns all functions with the specified omitted values, including functions subordinate to a universal covering map, rather than only the covering map itself. [S2]

Short English statement

Find image or omitted-value conditions forcing Taylor coefficients to tend to zero, including the proposed vanishing image-disc radius condition.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305007 (AMR-022-5007). 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 5.7, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.