Research Problems in Function Theory — Problem 5.42

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 $0$.Let $W^+$ be the analytic Wiener algebra: $f(z)=\sum_{n\ge0}a_nz^n$ with $\|f\|_{W^+}=\sum_{n\ge0}|a_n|<\infty$. For $0<\delta\le1$, define \[V(\delta)=\sup\left\{\sum_{n\ge0}|b_n|:\|f\|_{W^+}=1,\ |f(z)|\ge\delta\ (z\in\mathbb D),\ \frac1{f(z)}=\sum_{n\ge0}b_nz^n\right\}, \] allowing the supremum to be $\infty$. Set $\delta_* =\inf\{\delta\in(0,1]:V(\delta)<\infty\}$. The bound in $V(\delta)$ is uniform over all admissible $f$ and may depend only on $\delta$.

Statement recorded in the supplied source.

Determine $\delta_*$. In particular, is $\delta_*=1/2$? [S2]

Short English statement

Determine the smallest lower-modulus threshold above which normalized absolutely convergent power series have a uniform absolute-coefficient bound for their reciprocals.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305042 (AMR-022-5042). 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 5.42, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.