Research Problems in Function Theory — Problem 6.21

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 is typically real if $f(z)\in\mathbb R$ exactly when $z\in\mathbb R$, for $z\in\mathbb D$. Consider normalized typically-real functions $f(0)=0$, $f'(0)=1$, written $f(z)=zP(z)/(1-z^2)$, with $P$ holomorphic, $\operatorname{Re}P(z)>0$ and $P(0)=1$. Since $f$ is typically real, $P(\overline z)=\overline{P(z)}$.

Statement recorded in the supplied source.

Characterize the additional conditions on $P$ necessary and sufficient for $f(z)=zP(z)/(1-z^2)$ to be univalent on $\mathbb D$. [S2]

Short English statement

Which positive-real-part factors in the typically-real representation yield a univalent function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306021 (AMR-022-6021). 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 6, Problem 6.21, complete update and preceding notation. https://arxiv.org/abs/1809.07200.