Research Problems in Function Theory — Problem 5.2

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.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and let $f(z)=\sum_{n=0}^\infty a_nz^n$ be holomorphic in $\mathbb D$. For $\lambda>0$ and $0<r<1$ define the integral mean \[I_\lambda(r,f)=\left(\frac1{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^\lambda\,d\theta\right)^{1/\lambda}. \] The coefficient estimates refer to $n\to\infty$, and mean estimates to $r\uparrow1$. Constants in $O(\cdot)$ may depend on $f$ and the indicated exponent, but not on $n$ or $r$. Suppose there is a sequence of omitted values $w_k\in\mathbb C\setminus\{0\}$ such that $f(z)\ne w_k$ for every $z\in\mathbb D$, and their moduli $r_k=|w_k|$ satisfy $0<r_1\le r_2\le\cdots$ and $r_k\to\infty$. Equal successive moduli may be removed without changing the asymptotic hypotheses; any finitely many initial zero omitted values play no role. Assume the square-log summability condition \[\sum_{k=1}^\infty\left(\log\frac{r_{k+1}}{r_k}\right)^2<\infty. \]

Statement recorded in the supplied source.

Must both estimates \[I_1(r,f)=O\left((1-r)^{-1}\right)\quad(r\uparrow1),\qquad |a_n|=O(n)\quad(n\to\infty) \] hold for every such $f$? [S2]

Short English statement

Does summability of the squared logarithmic gaps between omitted-value moduli give a first mean of order at most one over the boundary distance and coefficients of order at most n?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305002 (AMR-022-5002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Chapter 5 introduction, equations (5.1)--(5.6), Problem 5.2, and complete Updates 5.1--5.3. https://arxiv.org/pdf/1809.07200.