Research Problems in Function Theory — Problem 6.14′

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 a mean $p$-valent function $f(z)=\sum_{j\geq0}a_jz^j$ and an integer $k\geq1$, form its coefficient Hankel determinant \[A_n^{(k)}(f)=\det(a_{n+i+j})_{0\leq i,j<k}.\] An exponent $a$ is admissible for $(p,k)$ if $|A_n^{(k)}(f)|^{1/k}=O_f(n^a)$ as $n\to\infty$ for every mean $p$-valent $f$; its bound can depend on $f,p,k$. Let $j_k(p)$ be the infimum of admissible exponents.

Statement recorded in the supplied source.

Determine the best possible exponents $j_k(p)$ for the mean $p$-valent class and whether the corresponding boundary bounds hold. The source extends equation(6.6) to this class with the admissible exponent \[-\frac12+16\sqrt{\frac{p^3}{k}},\] and asks for the sharp improvement. The normalization $\mu_p=1$ is not required for the individual-function asymptotic estimate. [S2]

Short English statement

Find the sharp power growth of coefficient Hankel determinants for mean-valent functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306814 (AMR-022-6814). 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 6.14 prime, complete problem and update, Problem6.14, determinant definition and equation(6.6), mean-p notation and update. https://arxiv.org/abs/1809.07200.