Research Problems in Function Theory — Problem 6.68

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 $f(z)=z+b_0+\sum_{j=1}^{\infty}b_jz^{-j}$. For locally injective holomorphic $f$, put $P_f=f''/f'$ and $S_f=P_f'-\tfrac12P_f^2$, its pre-Schwarzian and Schwarzian derivatives.

Statement recorded in the supplied source.

For every $f\in\Sigma$ and sequence $(f_n)\subset\Sigma$, does \[\sup_{|z|>1}(|z|^2-1)^2|S_{f_n}(z)-S_f(z)|\longrightarrow0\] imply \[\sup_{|z|>1}(|z|^2-1)|P_{f_n}(z)-P_f(z)|\longrightarrow0?\] [S2]

Short English statement

Does convergence of exterior Schwarzians in their weighted uniform norm imply convergence of pre-Schwarzians in the corresponding weaker norm?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306068 (AMR-022-6068). 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 6.68 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.