Research Problems in Function Theory — Problem 5.77

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. The disc algebra $A(\overline{\mathbb D})$ consists of the functions continuous on $\overline{\mathbb D}$ and analytic in $\mathbb D$. A univalent function means an injective analytic function. Let $m:[0,2\pi)\to\mathbb C$ be an arbitrary Lebesgue-measurable function.

Statement recorded in the supplied source.

Does every such $m$ occur as \[\lim_{r\uparrow1}f^{\prime}(re^{i\theta})=m(\theta)\quad\text{for almost every }\theta\] for some $f\in A(\overline{\mathbb D})$ that is univalent on $\mathbb D$? [S2]

Short English statement

Can every measurable complex boundary function be realized almost everywhere as the radial limit of the derivative of a univalent disc-algebra function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305077 (AMR-022-5077). 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 5.77, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.