Research Problems in Function Theory — Problem 6.111

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 $S$ be the class of holomorphic injective functions $f:\mathbb D\to\mathbb C$ with $f(0)=0$ and $f'(0)=1$. Let $\mathcal A=\{f\in\operatorname{Hol}(\mathbb D):f(0)=0,f'(0)=1\}$. If $f(z)=z+\sum_{k\geq2}a_kz^k$, $g(z)=z+\sum_{k\geq2}b_kz^k$, define \[N_\delta(f)=\{g\in\mathcal A:\sum_{k=2}^{\infty}k|a_k-b_k|\leq\delta\}.\] More generally a weight sequence $T_k\geq0$ gives $TN_\delta(f)$ by replacing $k$ with $T_k$. Let $St\subset S$ be the maps whose images are starlike about zero, meaning that every segment from zero to an image point stays in the image. For holomorphic $u,v$, $u\prec v$ means $u=v\circ\eta$ for a holomorphic $\eta:\mathbb D\to\mathbb D$ with $\eta(0)=0$. For real $-1\leq B<A\leq1$, define \[K[A,B]=\left\{f\in S:1+\frac{zf''(z)}{f'(z)}\prec\frac{1+Az}{1+Bz}\right\},\qquad \delta(A,B)=\begin{cases}(1-B)^{(A-B)/B},&B\ne0,\\e^{-A},&B=0.\end{cases}\] Define the largest uniform allowable radius by \[\delta_*(A,B)=\sup\{d\geq0:N_d(f)\subseteq St\text{ for every }f\in K[A,B]\}.\]

Statement recorded in the supplied source.

For every $-1\leq B<A\leq1$, do both $N_{\delta(A,B)}(f)\subseteq St$ for every $f\in K[A,B]$ and $\delta_*(A,B)=\delta(A,B)$ hold, without restricting to either \[-\tfrac14(\sqrt3+2)\leq B<A\leq1\quad\text{or}\quad -1\leq-A\leq B<A\leq1?\] [S2]

Short English statement

Does the known sharp neighborhood radius remain valid for every admissible Janowski parameter pair?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306111 (AMR-022-6111). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.111, complete problem and update. https://arxiv.org/abs/1809.07200.