Research Problems in Function Theory — Problem 6.12

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 $a_1=1$. Let the nonzero Taylor terms after $z$ occur at strictly increasing integer exponents $n_k\ge2$. The source gives the following implications: \[\liminf_{k\to\infty}\frac{n_{k+1}}{n_k}>1\ \Longrightarrow\ a_n=o(n^{-1}),\qquad \liminf_{k\to\infty}(n_{k+1}-n_k)>4\ \Longrightarrow\ a_n=o(n^{-1/2}).\] Coefficients outside the support are zero.

Statement recorded in the supplied source.

Are there intermediate conditions on the exponent gaps that yield coefficient-decay estimates between the two displayed implications? Find such conditions and estimates. [S2]

Short English statement

Find gap conditions between additive separation and Hadamard separation that give intermediate coefficient-decay estimates for normalized univalent functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306012 (AMR-022-6012). 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 6.12, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.