Research Problems in Function Theory — Problem 8.11

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 $\mathbb T=\partial\mathbb D$. The Dirichlet space $\mathcal D$ consists of analytic functions on $\mathbb D$ with \[\|h\|_{\mathcal D}^2=|h(0)|^2+\int_{\mathbb D}|h^{\prime}(z)|^2\,dA(z)<\infty.\] For $h\in\mathcal D$, let $\mathcal Ph=\{ph:p\in\mathbb C[z]\}$, and say $h$ is cyclic if its closure in this norm is all of $\mathcal D$.

Statement recorded in the supplied source.

If $g\in\mathcal D$ is cyclic, $f\in\mathcal D$, and $|f(z)|\ge|g(z)|$ for every $z\in\mathbb D$, must $f$ also be cyclic? [S2]

Short English statement

Does pointwise modulus domination of a cyclic Dirichlet function preserve cyclicity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308011 (AMR-022-8011). 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 8.11, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.