Research Problems in Function Theory — Problem 6.37

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $S$ consist of holomorphic injective functions $f(z)=z+\sum_{n\ge2}a_nz^n$ on $\mathbb D$; set $a_0=0$, $a_1=1$. A function in $S$ is starlike if its image is starlike about $0$, and convex if its image is a convex domain. Suppose $f$ is odd: $f(z)=z+\sum_{n\ge1}c_{2n+1}z^{2n+1}$, with $c_1=1$. Put $d_n=|c_{2n+1}|-|c_{2n-1}|$ for $n\ge1$; this is a signed real difference. An assertion $d_n=O(n^{-s})$ means $|d_n|\le C_f n^{-s}$ for all sufficiently large $n$.

Statement recorded in the supplied source.

For every odd $f\in S$, must $d_n=O(n^{-1/2})$ as $n\to\infty$? [S2]

Short English statement

Do the signed differences of successive odd coefficient moduli decay at the sharp inverse-square-root rate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306037 (AMR-022-6037). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 6, Problem 6.37, complete update and preceding notation. https://arxiv.org/abs/1809.07200.