Research Problems in Function Theory — Problem 5.76

Definitions and mathematical statement

Let $\mathbb D=\{z:|z|<1\}$ and $H^\infty(\mathbb D)=\{g\text{ holomorphic on }\mathbb D:\sup_{\mathbb D}|g|<\infty\}$. Let $\gamma$ be an arc with one endpoint $1$, otherwise contained in $\mathbb D$. The natural rotation convention is $\gamma_\theta=e^{i\theta}\gamma$, an arc ending at $e^{i\theta}$. With this convention, the requested property is \[ \exists g\in H^\infty(\mathbb D)\quad\forall\theta\in[0,2\pi):\qquad \lim_{z\to e^{i\theta},\ z\in\gamma_\theta}g(z)\text{ does not exist}. \] The source instead prints $\gamma_\theta=e^{i\theta\gamma}$. This exponential-of-an-arc expression is not the same as rotation and need not be an arc inside the disk; the rotation formula above is an explicit possible correction, not a silent transcription change.

There is a second substantive issue: the source calls $\gamma$ nontangential. In the usual Stolz-cone sense this means that, near $1$, \[ |1-z|\le C(1-|z|)\qquad(z\in\gamma) \] for some finite $C$. Under the rotation reading, every rotated arc is then a nontangential approach at its endpoint. The usual almost-everywhere nontangential-limit theorem for $H^\infty$ is incompatible with failure at every endpoint in that interpretation. Thus the intended arc hypothesis or approach convention must be checked; no alternate class is guessed.

The source also states that the tangential-arc case has an affirmative answer, citing Collingwood--Lohwater. This comparison and its precise tangential convention are retained as source claims. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no recent resolution located, but does not address the formula or nontangential-limit inconsistency.

Short English statement

Can a bounded analytic function fail to have a limit along every rotated copy of a prescribed approach arc? The printed rotation formula is malformed, and the standard nontangential reading conflicts with almost-everywhere boundary limits.

Sources

• Local UnsolvedMath registry, permanent ID $2305076$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 5.76."
• Hayman, Research Problems in Function Theory (2018), Problem 5.76; attributed to W. Rudin and communicated by K. F. Barth. https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• The source mentions Rudin and Collingwood--Lohwater without complete references. The supplied OPEN-TRIAGE summary gives no separate current-status citation.