Research Problems in Function Theory — Problem 6.84

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 $\mathbb T=\partial\mathbb D$. The class $S$ consists of holomorphic injective functions $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f'(0)=1$, so $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$. The holomorphic logarithm $\log(f(z)/z)$ is chosen to be $0$ at $z=0$, with the removable value $f(z)/z=1$ there. Define $\gamma_n$ and the normalized analytic root transforms by \[\log(f(z)/z)=2\sum_{n=1}^{\infty}\gamma_nz^n,\qquad F_p(z)=z\exp\!\left(\frac1p\log\frac{f(z^p)}{z^p}\right)=z+\sum_{n=1}^{\infty}c_n^{(p)}z^{pn+1}.\] Here $p\in\mathbb N$, and a coefficient $O$ bound may depend on $f$ and $p$.

Statement recorded in the supplied source.

If a fixed $f\in S$ satisfies $c_n^{(p)}=O(n^{2/p-1})$ as $n\to\infty$ for every integer $p\ge1$, must $\gamma_n=O(1/n)$? This asks for the converse of Milin's recorded implication from the logarithmic-coefficient bound to all root-transform bounds. [S2] Update 6.84 adds the universal $p=4$ root-transform bound to the known $p=1,2,3$ cases, while the original records failure for sufficiently large $p$. [S2]

Short English statement

Does having all the Szegő root-transform coefficient bounds force the logarithmic coefficients to decay at least as fast as one over their index?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306084 (AMR-022-6084). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.84 and its complete update; Chapter 6 initial notation. https://arxiv.org/abs/1809.07200.