Research Problems in Function Theory — Problem 6.5

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 $\Sigma$ be the class of analytic injective functions on $\mathbb D^*$ with Laurent expansion $F(z)=z+\sum_{n=1}^{\infty}b_nz^{-n}$. The area theorem gives $\sum_{n\ge1}n|b_n|^2\le1$. The cited estimates give $b_n=O(n^{-1/2-1/300})$, whereas examples give $|b_n|>n^{-1+\delta}$ infinitely often for one fixed $F$ and some absolute $\delta>0$. The update gives the lower example $|b_n|\ge n^{-0.83}$ infinitely often.

Statement recorded in the supplied source.

Determine the sharp orders of decay of the coefficients $b_n$ over $\Sigma$: find optimal general upper estimates as $n\to\infty$ and corresponding lower examples, with a single fixed $F$ where the source specifies infinitely many indices. In particular determine the optimal decay exponent and any finer factors needed for sharp magnitude estimates. [S2]

Short English statement

Determine the sharp decay rates of Laurent coefficients of normalized univalent functions outside the disk, including matching examples for a fixed function.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306005 (AMR-022-6005). 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 6.5, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.