Research Problems in Function Theory — Problem 5.17

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 $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$. For a plane domain $D$, write $\ell_D(r)$ for the angular Lebesgue measure of $\{\theta:re^{i\theta}\in D\}$. Its circular symmetrization $D^*$ contains the whole circle of radius $r$ when $D$ does; contains no points on that circle when $D$ misses it; otherwise it contains the arc $\{re^{i\theta}:|\theta|<\ell_D(r)/2\}$. Include $0$ in $D^*$ exactly when $0\in D$. Let $a_0>0$ belong to $D$, let $f:\mathbb D\to D$ be holomorphic with $f(0)=a_0$, and suppose $D^*$ admits a universal covering map $f^*:\mathbb D\to D^*$. Normalize the covering map by $f^*(0)=a_0$ and $(f^*)'(0)>0$. Write $f(z)=a_0+\sum_{n\ge1}a_nz^n$ and $f^*(z)=a_0+\sum_{n\ge1}a_n^*z^n$.Suppose $D$ and $D^*$ admit Green functions with pole at $a_0$, denoted $g_D(w,a_0)$ and $g_{D^*}(w,a_0)$. A Green function is the positive harmonic function away from its pole, with singularity $-\log|w-a_0|+O(1)$ there and zero boundary values in the Green-function sense. For $t>0$ set $D_t=\{w\in D:g_D(w,a_0)>t\}$ and $(D^*)_t=\{w\in D^*:g_{D^*}(w,a_0)>t\}$. Symmetrization of $D_t$ is taken about the same positive real ray. The slit plane is $\mathbb C\setminus(-\infty,0]$; the slit disc is $\mathbb D\setminus(-1,0]$.

Statement recorded in the supplied source.

Determine when the level-set inclusion \[(D_t)^*\subseteq(D^*)_t\qquad(t>0)\] holds. In particular, investigate the cases where $D^*$ is a slit plane or a slit disc. In the same general covering-map setup, determine when the coefficient domination \[|a_n|\le |a_n^*|\qquad\text{for every integer }n\ge1\] holds for all holomorphic maps $f$ into $D$ with $f(0)=a_0$. [S2]

Short English statement

Find domains for which symmetrization dominates Green-function level sets, especially slit domains, and determine when it dominates every Taylor coefficient.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305017 (AMR-022-5017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.17, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.