Research Problems in Function Theory — Problem 5.30

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 $0$. Let $S$ be the class of holomorphic injective functions $g:\mathbb D\to\mathbb C$ with $g(0)=0$ and $g'(0)=1$. Let $\mathcal B$ be the Bloch space of holomorphic functions $h$ with \[\|h\|_{\mathcal B}=|h(0)|+\sup_{z\in\mathbb D}(1-|z|^2)|h'(z)|<\infty. \] For $g\in S$, take the unique holomorphic logarithm $h=\log g'$ with $h(0)=0$, and let $\mathcal B_S=\{\log g':g\in S\}$. Let $\mathcal B_Q=\{\log g':g\in S\text{ admits a quasiconformal extension to }\widehat{\mathbb C}\}$. Such an extension is an orientation-preserving homeomorphism of the Riemann sphere that is locally in $W^{1,2}$ in charts and satisfies $|\partial_{\bar z}G|\le k|\partial_zG|$ almost everywhere for some $k<1$.

Statement recorded in the supplied source.

Is $\mathcal B_S$ connected in the Bloch norm topology? Is $\mathcal B_Q$ dense in $\mathcal B_S$ for that norm? Also determine geometrically its closure $\overline{\mathcal B_Q}^{\|\cdot\|_{\mathcal B}}$: give necessary and sufficient geometric conditions on a normalized univalent map $g$ for $\log g'$ to belong to that closure. [S2]

Short English statement

Study connectedness of the logarithmic-derivative class, density of the quasiconformally extendible subclass, and a geometric description of its closure.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305030 (AMR-022-5030). 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 5.30, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.