Research Problems in Function Theory — Problem 6.43

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.Use the holomorphic logarithm $\log(f(z)/z)$ with value $0$ at $z=0$, and define logarithmic coefficients by $\log(f(z)/z)=2\sum_{k\ge1}\gamma_kz^k$.

Statement recorded in the supplied source.

For every $f\in S$, does $\sum_{k\ge1}k|\gamma_k|r^k=O((1-r)^{-1})$ as $r\uparrow1$? The implied constant may depend on $f$. [S2]

Short English statement

Does the absolute logarithmic-coefficient sum have the same boundary growth order as the corresponding analytic derivative expression?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306043 (AMR-022-6043). 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 6.43, complete update and preceding notation. https://arxiv.org/abs/1809.07200.