Research Problems in Function Theory — Problem 6.79

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 $\mathbb T=\partial\mathbb D$. The class $S$ consists of holomorphic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f'(0)=1$, so $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. 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 $k<1$. For a fixed $0<k<1$, let $S_k(\infty)$ contain those $f\in S$ admitting a whole-plane quasiconformal extension $F$ with $|F_{\bar z}|\le k|F_z|$ almost everywhere and $F(\infty)=\infty$ on the sphere. Here $k$ bounds the Beltrami coefficient, rather than the dilatation $(1+k)/(1-k)$.

Statement recorded in the supplied source.

For every $f\in S_k(\infty)$ and $0<r<1$, must the normalized dilation $f_r(z)=f(rz)/r$ also belong to $S_k(\infty)$, with the same bound $k$? [S2] Update 6.79 attributes an affirmative answer to Krushkal. [S2]

Short English statement

Does restricting and rescaling a normalized univalent map preserve its exact quasiconformal extension bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306079 (AMR-022-6079). 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, Problem 6.79 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.