Research Problems in Function Theory — Problem 6.11

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 $S$ be the class of analytic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$, $f^{\prime}(0)=1$, with expansion $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$: $tw\in f(\mathbb D)$ for every $w\in f(\mathbb D)$ and $0\le t\le1$. Call $f\in S$ convex when $f(\mathbb D)$ is a convex domain. For convex $f,g\in S$ and $0<\lambda<1$, put $h=\lambda f+(1-\lambda)g$.

Statement recorded in the supplied source.

Is $h$ necessarily univalent and starlike about $0$? The original update reports a counterexample by Macgregor, together with the largest disk on which the starlikeness conclusion holds. [S2]

Short English statement

Does every convex combination of two normalized convex disk mappings remain univalent and starlike? The source records a counterexample.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306011 (AMR-022-6011). 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 6.11, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.