Research Problems in Function Theory — Problem 5.61

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 $w$ be analytic on $\mathbb D$ with $w(0)=0$. For $n\ge1$ define the differential operator \[P_nw(z)=\sum_{j=0}^{n}z^jw^{(j)}(z),\qquad w^{(0)}=w.\]

Statement recorded in the supplied source.

For every integer $n\ge1$, does $|P_nw(z)|<1$ for all $z\in\mathbb D$ imply $|w(z)|<1$ for all $z\in\mathbb D$? The source update reports an affirmative proof by Goldstein, Hall, Sheil-Small and Smith. [S2]

Short English statement

Record the higher-order differential-subordination question: does the stated differential sum mapping into the disk force w itself to map into the disk? The source reports that it was proved.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305061 (AMR-022-5061). 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 5.61, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.