Research Problems in Function Theory — Problem 9.3

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 let $H^{\infty}(D)$ denote the algebra of bounded analytic functions on a planar domain $D$, with supremum norm. Write $H^{\infty}=H^{\infty}(\mathbb D)$ and \[I(f_1,\ldots,f_n)=\left\{\sum_{j=1}^nf_jh_j:h_j\in H^{\infty}\right\}.\]

Statement recorded in the supplied source.

(a) For every finite $n$ and $f,f_1,\ldots,f_n\in H^{\infty}$, does $|f|\le\sum_{j=1}^n|f_j|$ throughout $\mathbb D$ imply $f\in I(f_1,\ldots,f_n)$? (b) For every pair $f_1,f_2\in H^{\infty}$, do there exist $f\in H^{\infty}$ and $\delta>0$ satisfying \[\delta(|f_1|+|f_2|)\le|f|\le|f_1|+|f_2|\quad\text{on }\mathbb D?\] When such a comparable $f$ exists, must it belong to $I(f_1,f_2)$, or can one at least choose one that does? The complete updates give negative examples to the two primary universal assertions. [S2]

Short English statement

Record the ideal-membership and comparable-modulus questions, including whether a comparable function must or may be chosen in the generated ideal. The source provides counterexamples to the universal assertions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309003 (AMR-022-9003). 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 9.3, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.