Research Problems in Function Theory — Problem 5.36

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 $1\le n_0<n_1<\cdots$ be integers and let $f(z)=\sum_{k\ge0}c_kz^{n_k}$ be holomorphic on $\mathbb D$, with $\sum_{k\ge0}|c_k|=\infty$. For real $\alpha<\beta$ with $0<\beta-\alpha\le2\pi$, let $V_{\alpha,\beta}=\{re^{i\theta}:0<r<1,\ \alpha<\theta<\beta\}$ be an open angular sector.Assume Hadamard gaps: there is a fixed $q>1$ with $n_{k+1}>q n_k$ for every $k\ge0$.

Statement recorded in the supplied source.

For every finite value $a\in\mathbb C$, must (i) $f(z)=a$ have at least one solution in $\mathbb D$; (ii) it have infinitely many distinct solutions in $\mathbb D$; and (iii) it have at least one solution in every sector $V_{\alpha,\beta}$? These are three separate assertions. [S2]

Short English statement

For a Hadamard-gap power series whose coefficients are not absolutely summable, ask separately whether it takes every finite value once, infinitely often, and in every angular sector.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305036 (AMR-022-5036). 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 5.36, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.