Research Problems in Function Theory — Problem 5.19

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.A meromorphic function $f$ on $\mathbb D$ has an asymptotic value $a\in\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ if $f(\gamma(t))\to a$ in the spherical metric along a continuous path $\gamma:[0,1)\to\mathbb D$ with $|\gamma(t)|\to1$. The path is not required to converge to a single boundary point. A value $a$ is locally exceptional at $\zeta\in\mathbb T$ if, in some neighborhood of $\zeta$ in $\mathbb D$, it is assumed only finitely many times; after shrinking the neighborhood it is omitted there.

Statement recorded in the supplied source.

Does there exist a meromorphic function $f$ on $\mathbb D$ with no asymptotic values, but with a locally exceptional value at some $\zeta\in\mathbb T$? [S2]

Short English statement

Historical question: can a meromorphic disc function without any asymptotic value still omit a value near one boundary point? The source update reports examples.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305019 (AMR-022-5019). 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 5.19, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.