Research Problems in Function Theory — Problem 5.40

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 $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.For holomorphic functions on $\mathbb D$, write $f\prec F$ when $f=F\circ\phi$ for a holomorphic map $\phi:\mathbb D\to\mathbb D$ satisfying $\phi(0)=0$. A Bloch function has $\sup_{z\in\mathbb D}(1-|z|^2)|F'(z)|<\infty$. Write $P(H)$ for \[\int_0^{2\pi}|H'(re^{i\theta})|^2\,d\theta=o((1-r)^{-2})\qquad(r\uparrow1),\] where little $o$ means that multiplication by $(1-r)^2$ gives a quantity tending to $0$.

Statement recorded in the supplied source.

Determine nontrivial conditions on $F$ that ensure $a_n\to0$ for every subordinate $f(z)=\sum_{n\ge0}a_nz^n\prec F$. In particular, is the conjunction of $F$ being Bloch and $P(F)$ sufficient? Separately, is $P$ preserved by subordination, that is, does $P(F)$ imply $P(f)$ for every $f\prec F$? [S2]

Short English statement

Find conditions on a dominating function forcing coefficient decay for all subordinates, including the Bloch and derivative-mean test, and ask whether that derivative-mean test survives subordination.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305040 (AMR-022-5040). 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 5, Problem 5.40, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.