Research Problems in Function Theory — Problem 6.14

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 $k,n\ge1$ define the Hankel determinant \[A_n^{(k)}=\det(a_{n+i+j})_{0\le i,j<k}.\] An exponent $\beta$ is admissible for order $k$ if for every $f\in S$ one has $|A_n^{(k)}|^{1/k}=O_f(n^{\beta})$ as $n\to\infty$. A function in $S$ is starlike if its image is starlike about $0$: $tw\in f(\mathbb D)$ for every $w\in f(\mathbb D)$ and $0\le t\le1$.

Statement recorded in the supplied source.

Are there admissible exponents strictly below $-1/2$ for all sufficiently large $k$? For the starlike subclass the source gives the sharp exponent $-1+2/k$. The update records $A_n^{(k)}=O(n^{3/2-k/2-k\beta_0})$ for $k\ge2$ and a constant $\beta_0>1/4000$, yielding an affirmative large-order conclusion. [S2]

Short English statement

Do Hankel determinants of univalent-function coefficients admit exponents below minus one half for all sufficiently large determinant orders? The source update answers the large-order question affirmatively.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306014 (AMR-022-6014). 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 6.14, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.