Research Problems in Function Theory — Problem 6.67

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 $\mathbb T=\partial\mathbb D$. For locally injective holomorphic $f$, put $P_f=f''/f'$ and $S_f=P_f'-\tfrac12P_f^2$, its pre-Schwarzian and Schwarzian derivatives. 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 $f$ be holomorphic and injective on $\mathbb D$ and assume $f(\mathbb D)$ is a Jordan domain. A quasiconformal extension across $\mathbb T$ extends $f$ to a quasiconformal homeomorphism on a neighbourhood of $\overline{\mathbb D}$. The limits below range over all points approaching the boundary, not only one radius.

Statement recorded in the supplied source.

Is each of the following conditions, separately, sufficient for such an extension? \[\limsup_{|z|\to1}(1-|z|^2)|P_f(z)|<2;\qquad \limsup_{|z|\to1}(1-|z|^2)^2|S_f(z)|<2.\] The source notes that the two conditions together suffice and that the constant $2$ would be sharp in each separate assertion. [S2] Update 6.67 reports that Gehring and Pommerenke answered the first, pre-Schwarzian question affirmatively. [S2]

Short English statement

Do either of two sharp boundary derivative bounds independently guarantee a quasiconformal extension of a Jordan conformal map?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306067 (AMR-022-6067). 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 6.67 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.