Research Problems in Function Theory — Problem 7.79

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 $G\subset\mathbb C$ be a domain and $f$ holomorphic there. Define $M(f)$ to consist of points with a neighborhood $N\subset G$ on which $\{f^{(n)}:n\ge1\}$ is normal. Normality means that every sequence has a subsequence converging locally uniformly in the spherical metric, with holomorphic/meromorphic limit or identically infinity as appropriate. Define $M_!(f)$ in the same way using $\{f^{(n)}/n!:n\ge1\}$. For meromorphic $f$ use the same spherical convention. A set is simply connected here if it is connected and every loop contracts within it.

Statement recorded in the supplied source.

Characterize the possible sets $M(f)$ beyond their being open. Must $M(f)$ be connected? If $G$ is simply connected, must $M(f)$ be simply connected? Can it be an arbitrary prescribed open subset of $G$? Answer the analogous questions for meromorphic $f$, and for the factorial-normalized derivative family defining $M_!(f)$. [S2]

Short English statement

Determine which open sets can be the region where a function’s derivatives form a normal family, including connectivity and the meromorphic and factorial-normalized variants.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307079 (AMR-022-7079). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.79, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.