Research Problems in Function Theory — Problem 6.13′

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 $\mu_p(f)=1$. Fix $p>1/4$ and an integer $m\geq1$. Call an exponent $a\in\mathbb R$ admissible if for every mean $p$-valent $f(z)=\sum_{j\geq0}a_jz^j$ and every integer $k\geq1$ there is $C=C(f,k,m,p)<\infty$ such that, for each sufficiently large integer $n$, one can choose complex numbers $c_0,\ldots,c_m$ with $c_0=1$, $|c_m|\geq4^{-m}$, and \[\left|\sum_{\nu=0}^{m}c_\nu a_{n+j+\nu}\right|\leq Cn^a\qquad(0\leq j\leq k).\] The coefficients $c_\nu$ may depend on $f,k,m,n$. Let $\alpha_m(p)$ be the infimum of admissible exponents. This definition applies to the full mean-valent class, without the normalization $\mu_p=1$; the bound may depend on $f$.

Statement recorded in the supplied source.

Determine the sharp exponents $\alpha_m(p)$ and whether each boundary exponent is admissible. In particular, are \[\alpha_1(p)=p-1\quad(1/2<p<1),\qquad \alpha_1(p)=-1/2\quad(1/4<p<1/2),\] and $\alpha_m(p)=-1/2$ whenever $m+1>4p$? The source reports the available admissible exponent $-1/2+8p^{3/2}/\sqrt m$ and the obstruction $\alpha_m(p)\geq2p/(m+1)-1$. [S2]

Short English statement

Determine the optimal powers controlling simultaneous finite linear combinations of consecutive coefficients, including the stated mean-valence conjectures.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306813 (AMR-022-6813). 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 6.13 prime, complete problem and update, Problem6.13, equation(6.5), complete mean-p notation, examples and update. https://arxiv.org/abs/1809.07200.