Research Problems in Function Theory — Problem 5.21

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 $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.For $w\in\mathbb D$ use the disc automorphism $\phi_w(z)=(z-w)/(1-\overline w z)$ and write \[f_w=f\circ\phi_w=\sum_{n=0}^\infty a_n^{(w)}z^n,\qquad W_f=\left\{w\in\mathbb D:\sum_{n=0}^\infty|a_n^{(w)}|<\infty\right\}. \] Here $f$ is any holomorphic function on $\mathbb D$. Positive measure means positive planar Lebesgue measure; dense means dense in $\mathbb D$.

Statement recorded in the supplied source.

Describe the possible sets $W_f$. Does each of the following assumptions, separately, imply $W_f=\mathbb D$: (i) $W_f$ is dense; (ii) $W_f$ is uncountable; (iii) $W_f$ has positive planar measure? If $f$ is univalent, must $W_f$ be either empty or all of $\mathbb D$? [S2]

Short English statement

Classify the centers at which a disc reparametrization has absolutely summable coefficients, including the three largeness criteria and the univalent case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305021 (AMR-022-5021). 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 5, Problem 5.21, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.