Research Problems in Function Theory — Problem 8.2

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 $H^\infty$ be bounded holomorphic functions on $\mathbb D$, and let $f_1,\ldots,f_N\in H^\infty$, $N\ge1$. Put $s(z)=\sum_{j=1}^N|f_j(z)|$, and define the algebraic ideals \[I=\{\sum_{j=1}^Nf_jh_j:h_j\in H^\infty\},\qquad J=\{g\in H^\infty: |g(z)|\le C_gs(z)\text{ on }\mathbb D \text{ for some }C_g<\infty\}. \] Thus $I\subset J$. The ideal $J^2$ consists of finite sums of products of two elements of $J$, not a norm closure.

Statement recorded in the supplied source.

Determine the relations between these ideals when they are proper. The originally printed exponent question asks whether an absolute $\kappa>0$ makes $|g(z)|^\kappa\le C s(z)$ imply $g\in I$; the source update explicitly replaces that formulation by the following question: find an absolute integer $k\ge2$ such that $g\in J$ implies $g^k\in I$. Retain separately the question $J^2\subset I$. The update records that $k=3$ works and is best possible; this corrected power-membership question is distinct from the initially printed pointwise-exponent formula. [S2]

Short English statement

Relate a bounded-function ideal to the functions controlled pointwise by its generators, including squared-ideal inclusion and the corrected universal power-membership question.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308002 (AMR-022-8002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 8.2, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.