Research Problems in Function Theory — Problem 2.73

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 $F:\mathbb C^3\to\mathbb C$ be entire jointly in its three variables. Say that its parameter dependence factors through one entire function if there are entire maps $G:\mathbb C^2\to\mathbb C$ and $H:\mathbb C^2\to\mathbb C$ with $F(z,a,b)=G(z,H(a,b))$ for all $z,a,b\in\mathbb C$. A family is normal if every sequence has a subsequence converging locally uniformly in the spherical metric to a meromorphic function or to $\infty$.

Statement recorded in the supplied source.

Does there exist such an entire function $F$ that does not admit this factorization and for which $\{z\mapsto F(z,a,b):(a,b)\in\mathbb C^2\}$ is a normal family on the whole plane? [S2]

Short English statement

Can there exist a normal entire family on the plane that has two genuine parameters and cannot be expressed through one entire scalar parameter?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302073 (AMR-022-2073). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.73. https://arxiv.org/abs/1809.07200.