Research Problems in Function Theory — Problem 6.58

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.

Let $\Sigma$ here consist of holomorphic injective exterior functions $f(z)=z+b_0+\sum_{n\ge1}b_nz^{-n}$ on $|z|>1$, as in Problem 6.57. The additive constant is unrestricted. A $q$-quasiconformal extension is an orientation-preserving homeomorphism of the Riemann sphere fixing $\infty$ (equivalently a homeomorphism of the whole plane extended at infinity), locally in $W^{1,2}$ in charts, agreeing with the function on its domain and satisfying $|\partial_{\bar z}F|\le q|\partial_zF|$ almost everywhere for $0\le q<1$.For distinct $z,\zeta$ with $|z|,|\zeta|>1$, put \[L_f(z,\zeta)=\log\frac{f'(z)f'(\zeta)(z-\zeta)^2}{(f(z)-f(\zeta))^2},\] using the holomorphic branch that extends to $0$ on the diagonal $z=\zeta$. The quotient is nonzero and equals $1$ on that diagonal. For $0<q<1$, consider the Goluzin condition \[|L_f(z,\zeta)|\le q\log\frac{|z\overline\zeta-1|^2}{(|z|^2-1)(|\zeta|^2-1)}\qquad(|z|,|\zeta|>1).\]

Statement recorded in the supplied source.

Is this condition sufficient for $f$ to admit a quasiconformal extension to the whole plane? Does sufficiency hold at least for every sufficiently small positive $q$? [S2]

Short English statement

Does the two-point Goluzin distortion inequality characterize quasiconformal extendibility of exterior univalent functions, at least for small distortion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306058 (AMR-022-6058). 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 6, Problem 6.58, complete update and preceding notation. https://arxiv.org/abs/1809.07200.