Research Problems in Function Theory — Problem 8.20

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $H^{\infty}(\mathbb D)$ be the algebra of bounded analytic functions, with norm $\|h\|_\infty=\sup_{\mathbb D}|h|$. For $n\ge1$ let $f_1,\ldots,f_n,g\in H^{\infty}(\mathbb D)$ with $|g(z)|\le\sum_{j=1}^n|f_j(z)|$ throughout $\mathbb D$. The ideal generated by the $f_j$ consists of sums $\sum_{j=1}^nf_jh_j$ with $h_j\in H^{\infty}$.

Statement recorded in the supplied source.

Must $g^2$ belong to this ideal, equivalently must there exist $h_1,\ldots,h_n\in H^{\infty}$ such that $g^2=\sum_{j=1}^nf_jh_j$? The source records a theorem for $g^3$ and counterexamples for $g$ itself. [S2]

Short English statement

Does the pointwise bound by a finite sum of generator moduli force the square into the bounded-analytic-function ideal?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308020 (AMR-022-8020). 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 8.20, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.