Research Problems in Function Theory — Problem 6.27

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 $g$ be holomorphic and injective on $\{|z|>1\}$, with Laurent expansion $g(z)=z+b_0+\sum_{n\ge1}b_nz^{-n}$. For integers $n\ge3$, define $H_n=\max\{\nu(|b_\nu|+1):\nu\ge1,\ 0<|\nu-n|<n/2\}$. This is the literal coefficient expression recorded in the source. The exterior area theorem gives $|b_n|\le n^{-1/2}$; also $H_n\ge n/2$, so this literal version already has an elementary bound.

Statement recorded in the supplied source.

For every $\varepsilon>0$, does there exist $C_{g,\varepsilon}<\infty$ such that $n|b_n|\le C_{g,\varepsilon}n^\varepsilon H_n$ for all sufficiently large $n$? [S2]

Short English statement

Record the exterior coefficient comparison as printed, including its neighboring-coefficient expression. That literal version follows from the area theorem.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306027 (AMR-022-6027). 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 6.27, complete update and preceding notation. https://arxiv.org/abs/1809.07200.