Research Problems in Function Theory — Problem 6.18

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$, $\mathbb T=\partial\mathbb D$, and $\mathbb D^*=\{z\in\mathbb C:|z|>1\}$. Let $\Sigma$ be the class of analytic injective functions on $\mathbb D^*$ with Laurent expansion $F(z)=z+\sum_{n=1}^{\infty}b_nz^{-n}$. The omitted compact set is $K_F=\mathbb C\setminus F(\mathbb D^*)$, and its area is \[A(F)=\operatorname{area}(K_F)=\pi\left(1-\sum_{n=1}^{\infty}n|b_n|^2\right).\] The Schwarzian derivative is \[\mathcal S_F(z)=\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{\prime}(z)-\frac12\left(\frac{F^{\prime\prime}(z)}{F^{\prime}(z)}\right)^2.\] Let $F_j,F\in\Sigma$ and assume $F_j(z)\to F(z)$ for every $|z|>1$.

Statement recorded in the supplied source.

Under what additional hypotheses does $A(F_j)\to A(F)$? In particular, find suitable hypotheses involving the weighted Schwarzian $(1-|z|^2)^2\mathcal S_F(z)$ and, where appropriate, those of the approximating functions. The source update flags errors in the original wording; the area here refers to the complement of the exterior image, as forced by the stated Laurent class and area formula. [S2]

Short English statement

Find conditions under which the areas of omitted sets converge for convergent exterior univalent functions, especially conditions involving weighted Schwarzians.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306018 (AMR-022-6018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 6.18, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.