Research Problems in Function Theory — Problem 9.15

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. Put $H^{\infty}=H^{\infty}(\mathbb D)$ and $I(f_1,f_2)=\{f_1h_1+f_2h_2:h_1,h_2\in H^{\infty}\}$. Let $f_1,f_2,g\in H^{\infty}$ satisfy $|g|\le|f_1|+|f_2|$ throughout $\mathbb D$.

Statement recorded in the supplied source.

Is $g^2\in I(f_1,f_2)$ necessarily true? Equivalently, must $g^2=f_1h_1+f_2h_2$ for some bounded analytic $h_1,h_2$? The source records membership for the cube and counterexamples for the first power. [S2]

Short English statement

Does modulus domination by two bounded analytic generators force the square into their ideal?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309015 (AMR-022-9015). 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 9.15, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.