Research Problems in Function Theory — Problem 6.101

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.

For a nonconstant holomorphic function $f$ on a domain $D$, its valence is $v_D(f)=\sup_{w\in\mathbb C}n_D(w,f)\in\mathbb N\cup\{\infty\}$, where $n_D(w,f)$ counts the zeros of $f-w$ in $D$ with multiplicity. The function is $p$-valent if $v_D(f)\leq p$. Let $K\subseteq\mathbb C$ be closed and $p\geq1$ an integer. Set \[\mathcal F(K)=\left\{z\longmapsto\sum_{k=1}^{m}\frac{A_k}{z-a_k}:m\geq1,\ A_k>0,\ a_k\in K\right\}.\] A common domain of $p$-valence is a nonempty connected open set $D\subseteq\mathbb C\setminus K$ on which $v_D(f)\leq p$ for every $f\in\mathcal F(K)$. Such a domain is maximal if it is contained in no strictly larger common domain of $p$-valence.

Statement recorded in the supplied source.

For a given closed set $K$ and integer $p>1$, determine a maximal common domain of $p$-valence for $\mathcal F(K)$. The source records that the case $p=1$ was solved by Distler. [S2]

Short English statement

Find a largest-by-inclusion common region of bounded valence for rational functions with positive residues at poles in a prescribed closed set.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306101 (AMR-022-6101). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.101, complete problem and update. https://arxiv.org/abs/1809.07200.