Research Problems in Function Theory — Problem 6.16

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 $0\le t\le1$. Write $S^*$ for the starlike subclass and $k(z)=z/(1-z)^2$ for the Koebe function, with $k^{\prime}(z)=(1+z)/(1-z)^3$.

Statement recorded in the supplied source.

For every $z_0\in\mathbb D$, does the equality of value regions \[\{f^{\prime}(z_0):f\in S^*\}=\{k^{\prime}(z):|z|\le|z_0|\}\] hold? The source update records Hummel’s counterexample to this version of the Marx conjecture. [S2]

Short English statement

Does the derivative value region for normalized starlike functions equal the Koebe derivative image of the corresponding closed disk? The source records a counterexample.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306016 (AMR-022-6016). 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 6.16, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.