Research Problems in Function Theory — Problem 6.52

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 $\mathbb T=\partial\mathbb D$.A bounded univalent function is a holomorphic injective $g:\mathbb D\to\mathbb C$ with $\sup_{z\in\mathbb D}|g(z)|<\infty$; no normalization of $g$ is imposed.

Statement recorded in the supplied source.

For every holomorphic surjection $f:\mathbb D\to\mathbb C$, does there exist such a $g$ for which $(f+g)(\mathbb D)=\mathbb C$? [S2]

Short English statement

Can every holomorphic disc surjection onto the plane be perturbed by a bounded injective function while retaining its full range?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306052 (AMR-022-6052). 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 6, Problem 6.52, complete update and preceding notation. https://arxiv.org/abs/1809.07200.