Research Problems in Function Theory — Problem 6.80

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 $\mathbb T=\partial\mathbb D$. A holomorphic function $h$ on $\mathbb D$ is normal if the family $\{h\circ\psi:\psi\in\operatorname{Aut}(\mathbb D)\}$ is normal in the spherical metric on compact subsets. Equivalently, $\sup_{z\in\mathbb D}(1-|z|^2)|h'(z)|/(1+|h(z)|^2)<\infty$. For holomorphic injective $f$ on $\mathbb D$, its primitive $F(z)=\int_0^z f(t)\,dt$ is independent of the integration path.

Statement recorded in the supplied source.

Must $F$ be a normal function for every holomorphic injective $f$ on $\mathbb D$? No normalization of $f$ at $0$ is assumed. [S2]

Short English statement

Is the first primitive of every univalent function in the disc a normal function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306080 (AMR-022-6080). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.80 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.