Research Problems in Function Theory — Problem 6.69

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.

The exterior class $\Sigma$ consists of holomorphic injective functions on $\{z:|z|>1\}$ with Laurent expansion $f(z)=z+b_0+\sum_{j=1}^{\infty}b_jz^{-j}$. A quasiconformal map is an orientation-preserving homeomorphism locally in $W^{1,2}$ whose distributional derivatives satisfy $|F_{\bar z}|\le k|F_z|$ almost everywhere for some $k<1$. Let $B$ be the vector space of holomorphic functions on $\{|z|>1\}$ with finite norm $\|\phi\|=\sup_{|z|>1}(|z|^2-1)|z\phi(z)|$. In this norm topology set $\mathcal S=\{f''/f':f\in\Sigma\}$ and $\mathcal T=\{f''/f':f\in\Sigma\text{ admits a quasiconformal extension to }\mathbb C\}$. The source's summation symbol in this set definition denotes the exterior class $\Sigma$ defined in Problem 6.68.

Statement recorded in the supplied source.

Is $\mathcal T=\operatorname{int}_B\mathcal S$? [S2] Update 6.69 attributes an affirmative answer to Astala and Gehring. [S2]

Short English statement

Are exactly the interior pre-Schwarzians of normalized exterior univalent maps those admitting quasiconformal extensions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306069 (AMR-022-6069). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problems 6.68 and 6.69, and Update 6.69. https://arxiv.org/abs/1809.07200.