Research Problems in Function Theory — Problem 6.10

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 $\mathbb D=\{z\in\mathbb C:|z|<1\}$, $\mathbb T=\partial\mathbb D$, and $\mathbb D^*=\{z\in\mathbb C:|z|>1\}$. Let $\Sigma$ be the class of analytic injective functions on $\mathbb D^*$ with Laurent expansion $F(z)=z+\sum_{n=1}^{\infty}b_nz^{-n}$. Call $F\in\Sigma$ convex if the omitted compact set $K_F=\mathbb C\setminus F(\mathbb D^*)$ is convex. For convex $F,G\in\Sigma$ and $0<\lambda<1$, put $H=\lambda F+(1-\lambda)G$. The source records that $H\in\Sigma$.

Statement recorded in the supplied source.

Is $K_H=\mathbb C\setminus H(\mathbb D^*)$ necessarily convex? [S2]

Short English statement

Does a convex combination of two exterior convex univalent functions retain a convex omitted set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306010 (AMR-022-6010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 6.10, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.