Research Problems in Function Theory — Problem 5.32

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 $k<1$. Let $h_n=\log g_n'\in\mathcal B_S$ and $h=\log g'\in\mathcal B_S$, with the logarithms normalized at $0$. Their Bloch-norm difference is $\|h_n-h\|_{\mathcal B}=\sup_{z\in\mathbb D}(1-|z|^2)|h_n'(z)-h'(z)|$.

Statement recorded in the supplied source.

Give a geometric characterization, in terms of the normalized conformal maps $g_n,g$ and their image domains, of the convergence $\|h_n-h\|_{\mathcal B}\to0$. Find conditions that are both necessary and sufficient, rather than only compact-local convergence of $g_n$. [S2]

Short English statement

Describe geometrically what convergence of logarithmic derivatives in the Bloch norm means for the corresponding normalized univalent maps.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305032 (AMR-022-5032). 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 5.32, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.