Research Problems in Function Theory — Problem 7.25

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 $K\subset\mathbb C$ be compact with positive planar Lebesgue measure. For $f$ holomorphic on $\mathbb C\setminus K$ and distinct $z,\zeta$ there, define \[Q_f(z,\zeta)=\frac{f(z)-f(\zeta)}{z-\zeta}.\] The condition $f(\infty)=0$ means $f(z)\to0$ as $|z|\to\infty$. A nonlinear function means one that is not the restriction of an affine function $az+b$.

Statement recorded in the supplied source.

(a) Must there exist a nonconstant such $f$, with $f(\infty)=0$, for which $Q_f(z,\zeta)\notin\{-1,1\}$ for all distinct $z,\zeta\in\mathbb C\setminus K$? (b) Does every such $K$ admit a nonlinear holomorphic function on $\mathbb C\setminus K$ with uniformly bounded difference quotients: \[\sup_{z\ne\zeta,\ z,\zeta\notin K}|Q_f(z,\zeta)|<\infty?\] (c) More generally, for a prescribed plane set $V\subset\mathbb C$, determine when there exists a nonconstant holomorphic $f$ with $f(\infty)=0$ whose difference quotients omit every value in $V$. [S2]

Short English statement

For a compact set of positive area, find holomorphic functions whose difference quotients avoid the two specified values, ask whether bounded difference quotients are possible, and determine which other prescribed sets of values can be omitted.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307025 (AMR-022-7025). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.25 and its complete update. https://arxiv.org/abs/1809.07200.