Research Problems in Function Theory — Problem 6.2′

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 $\mu_p(f)=1$. For each $p>0$ and integer $n\geq1$, define $A_n^{(p)}=\sup_{f\in S(p)}|a_n|$.

Statement recorded in the supplied source.

For each $p>0$, does a finite limit \[\lim_{n\to\infty}\frac{A_n^{(p)}}{n^{2p-1}}=K_p\] exist, and if it exists, what is its value? For comparison, the source reports existence of the corresponding limit for each fixed $f\in S(p)$ when $p>1/4$. [S2]

Short English statement

Does the extremal coefficient magnitude in the normalized mean-valent class have a limiting power-law constant?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306802 (AMR-022-6802). 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 6.2 prime, complete problem and update, preceding areal-mean-valence definition and coefficient normalization. https://arxiv.org/abs/1809.07200.