Research Problems in Function Theory — Problem 6.19

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, $\mathbb T=\partial\mathbb D$, and $\mathbb D^*=\{z\in\mathbb C:|z|>1\}$. Let $S$ be the class of analytic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$, $f^{\prime}(0)=1$, with expansion $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $a_1=1$. If $\sum_{n\ge1}|a_n|<\infty$, the Taylor series converges uniformly on $\overline{\mathbb D}$ and defines a continuous boundary function. For the more general alternative let $f(z)=\sum_{n\ge1}a_nz^n$ be analytic with the same absolute summability, without requiring the normalization or global injectivity of $S$.

Statement recorded in the supplied source.

Can the compact boundary image $f(\mathbb T)$ have positive two-dimensional Lebesgue measure (a) when $f\in S$, or (b) under the weaker requirement $f^{\prime}(z)\ne0$ for every $z\in\mathbb D$? [S2]

Short English statement

Can an absolutely summable analytic Taylor series have a boundary image of positive area, either for a normalized univalent function or merely a locally univalent one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306019 (AMR-022-6019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 6.19, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.