Research Problems in Function Theory — Problem 6.22

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.Let $S_{1/2}^*\subset S$ consist of functions with $\operatorname{Re}(zf'(z)/f(z))\ge1/2$, extending the quotient at $0$ by $1$. Such a function can be written with $P(z)=2zf'(z)/f(z)-1$, where $P(0)=1$ and $\operatorname{Re}P>0$. Define $R_{1/2}$ as the supremum of $R\in(0,1]$ such that every $f\in S_{1/2}^*$ maps $\{|z|<R\}$ onto a convex domain.

Statement recorded in the supplied source.

Determine $R_{1/2}$ sharply. Equivalently, determine the largest radius on which, for every holomorphic $P$ with $P(0)=1$ and $\operatorname{Re}P>0$, \[\operatorname{Re}\left(\frac{P(z)+1}{2}+\frac{zP'(z)}{P(z)+1}\right)>0.\] [S2]

Short English statement

Determine the universal radius of convexity for normalized functions starlike of order one half.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306022 (AMR-022-6022). 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 6.22, complete update and preceding notation. https://arxiv.org/abs/1809.07200.