Research Problems in Function Theory — Problem 6.3

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, $\mathbb T=\partial\mathbb D$, and $\mathbb D^*=\{z\in\mathbb C:|z|>1\}$. Let $S$ be the class of analytic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$, $f^{\prime}(0)=1$, with expansion $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $a_1=1$. For each $n\ge2$, call $c_n$ admissible if \[|\operatorname{Re}(n-a_n)|\le c_n\operatorname{Re}(2-a_2)\qquad\text{for every }f\in S.\] The constants are independent of $f$.

Statement recorded in the supplied source.

Determine the least possible $c_n$ for every $n\ge2$. Also, for each $n$, does a finite constant $d_n$, independent of $f$, exist such that \[\big|n-|a_n|\big|\le d_n(2-|a_2|)\qquad\text{for every }f\in S?\] [S2]

Short English statement

Find the sharp constants controlling a coefficient’s deviation from the Koebe value through the second coefficient, and determine whether the analogous modulus estimate has a finite constant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306003 (AMR-022-6003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 6.3, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.