Research Problems in Function Theory — Problem 8.25

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 $\mathbb T=\partial\mathbb D$. Let $A$ be the boundary restrictions to $\mathbb T$ of the disc algebra: functions continuous on $\overline{\mathbb D}$ and analytic in $\mathbb D$. Let $\psi:\mathbb T\to\mathbb T$ be an orientation-reversing homeomorphism and set \[A_\psi=\{f\in C(\mathbb T):f\in A\text{ and }f\circ\psi\in A\}.\]

Statement recorded in the supplied source.

Characterize the orientation-reversing homeomorphisms $\psi$ for which $A_\psi$ consists exactly of the constant functions. [S2]

Short English statement

When can a circle function and its orientation-reversed reparametrization both lie in the disc algebra only if the function is constant?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308025 (AMR-022-8025). 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 8.25, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.