Research Problems in Function Theory — Problem 6.7

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 $a_1=1$. Consider bounded members of $S$: for some finite $M$, $|f(z)|<M$ throughout $\mathbb D$. Then $\sum_{n\ge1}n|a_n|^2\le M^2$, and the source records $|a_n|=O(n^{-1/2-1/300})$. It also records examples with $|a_n|>n^{-1+\delta}$ at infinitely many indices for a fixed bounded $f$ and some $\delta>0$.

Statement recorded in the supplied source.

Determine the sharp coefficient-decay orders for bounded functions in $S$, with optimal general upper estimates and matching fixed-function lower examples. The update relates this question to exterior coefficient decay and reports the conjectured exponent $3/4$ in bounds of the form $n^{-\gamma+o(1)}$. [S2]

Short English statement

Determine the sharp decay rates of Taylor coefficients of bounded normalized univalent disk functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306007 (AMR-022-6007). 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 6.7, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.