Research Problems in Function Theory — Problem 6.33

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 $\mathbb T=\partial\mathbb D$. Let $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain.A slit is a non-self-intersecting piecewise analytic arc from a finite point $w_0$ to $\infty$. Boundary values at the end of the slit refer to the continuous spherical boundary extension of the slit-domain conformal map. For a map normalized by $f(-1)=\infty$, write $f(e^{i\phi_f})=w_0$, with $-\pi<\phi_f<\pi$.For $0<\alpha\le2$, let $T_\alpha$ consist of maps whose complement slit lies in some infinite sector of opening $\alpha\pi$ with vertex $w_0$. A sector is $\{w_0+re^{it}:r\ge0,\ |t-\theta|\le\alpha\pi/2\}$ for some $\theta\in\mathbb R$.

Statement recorded in the supplied source.

Determine sharp bounds on $|a_n|$ for all $n\ge2$ in $T_\alpha$, and determine $\sup|\phi_f|$ for maps in this class with $f(-1)=\infty$. [S2]

Short English statement

Find coefficient and slit-tip preimage-angle bounds when the omitted slit lies inside a sector based at its finite endpoint.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306033 (AMR-022-6033). 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 6.33, complete update and preceding notation. https://arxiv.org/abs/1809.07200.