Research Problems in Function Theory — Problem 5.13

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$. A family of meromorphic functions on $\mathbb D$ is normal if every sequence has a subsequence converging locally uniformly in the spherical metric to a meromorphic function or to the constant $\infty$. Differential inequalities below mean that the indicated meromorphic functions never take the stated finite value; poles do not violate a finite-value omission.

Statement recorded in the supplied source.

For each fixed integer $n\ge3$, is the family \[\mathcal G_n=\{f\text{ meromorphic on }\mathbb D:f'(z)f(z)^n\ne1\text{ for every }z\in\mathbb D\}\] normal on $\mathbb D$? [S2]

Short English statement

Historical question: does omission of one by the differential monomial $f'f^n$ ensure normality for meromorphic disc functions when $n$ is at least three? The source update reports an affirmative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305013 (AMR-022-5013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.13, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.