Research Problems in Function Theory — Problem 5.55

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 $\mathbb T=\partial\mathbb D$. For an analytic $h$ put $G_h=h(\mathbb D)$. Say $h$ has the $L$-property on a sequence $(z_j)$ with $|z_j|\to1$ if, for every compact $K\subset G_h$, only finitely many indices $j$ satisfy $h(z_j)\in K$. An ordered $L$-pair $(f,g)$ means that every such sequence on which $f$ has the $L$-property is also one on which $g$ has it. A nonconstant analytic $\alpha$ is an $L$-atom if, for every analytic $f$ for which $(f,\alpha)$ is an ordered $L$-pair, there is an analytic $\phi_f:G_\alpha\to\mathbb C$ with $f=\phi_f\circ\alpha$. For a general planar domain $D$, replace $|z_j|\to1$ by the requirement that $(z_j)$ eventually leaves every compact subset of $D$, and use the same range-compactness and factorization definitions.

Statement recorded in the supplied source.

Is $\alpha$ an $L$-atom if and only if $\alpha$ is univalent (injective)? Determine what becomes of this equivalence when the source domain is a general planar domain $D$. [S2]

Short English statement

Characterize the analytic functions that force every preceding member of an ordered escape pair to factor through them; are these exactly the univalent functions, in the disk and in other domains?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305055 (AMR-022-5055). 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 5.55, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.