Research Problems in Function Theory — Problem 6.31

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.Suppose for constants $\lambda\in\mathbb C\setminus\{0\}$ and $\delta>0$ that $(1-r)^2f(r)=\lambda+O((1-r)^\delta)$ as real $r\uparrow1$. The constants in all asymptotics may depend on $f,\lambda,\delta$.

Statement recorded in the supplied source.

Determine the sharp improvement, under exactly this hypothesis, of the estimate $a_n/n=\lambda+O(1/\log n)$ as $n\to\infty$. [S2]

Short English statement

How quickly must the normalized coefficients approach the nonzero radial growth limit when the radial error has a positive power rate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306031 (AMR-022-6031). 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 6.31, complete update and preceding notation. https://arxiv.org/abs/1809.07200.