Research Problems in Function Theory — Problem 6.7′

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 $\mathbb T=\partial\mathbb D$. For $p>0$ and a holomorphic $f$ on $\mathbb D$, let $n_f(w)$ count the preimages of $w$ with multiplicity. The function is areally mean $p$-valent when \[\int_0^{2\pi}\int_0^R n_f(\rho e^{i\theta})\rho\,d\rho\,d\theta\leq\pi pR^2\qquad(R>0).\] Writing $f(z)=\sum_{j=0}^{\infty}a_jz^j$, put $\mu_p(f)=\max\{|a_j|:j\in\mathbb Z,\ 0\leq j\leq p\}$, and let $S(p)$ be the mean $p$-valent functions with $\mu_p(f)=1$. Here bounded means $\sup_{z\in\mathbb D}|f(z)|<\infty$. The relation $u_n=o(n^{-1/2})$ means $n^{1/2}u_n\to0$, while $u_n=O(n^{-1/2-\varepsilon})$ means $n^{1/2+\varepsilon}|u_n|$ is eventually bounded.

Statement recorded in the supplied source.

Record the sharp decay assertion stated in the source: for every $p>0$ and bounded $f\in S(p)$, \[|a_n|=o(n^{-1/2}),\] and no fixed improvement of the exponent is valid throughout the class: for every $p>0$ and $\varepsilon>0$ there is a bounded $f\in S(p)$ whose coefficients are not $O(n^{-1/2-\varepsilon})$. The source presents this as a consequence of its preceding counterexample, rather than as a remaining conjecture. [S2]

Short English statement

Bounded mean-valent functions have little-o square-root coefficient decay, but no uniform improvement to a larger power exponent.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306807 (AMR-022-6807). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.7 prime, complete problem and update, complete preceding sparse-coefficient counterexample and normalization. https://arxiv.org/abs/1809.07200.