Research Problems in Function Theory — Problem 5.16

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 $f^*(z)=a_0+\sum_{n\ge1}a_n^*z^n$.Define, for $0<r<1$ and $\lambda>0$, \[I_\lambda(r,h)=\left(\frac1{2\pi}\int_0^{2\pi}|h(re^{i\theta})|^\lambda\,d\theta\right)^{1/\lambda},\qquad T(r,h)=\frac1{2\pi}\int_0^{2\pi}\log^+|h(re^{i\theta})|\,d\theta,\] where $\log^+x=\max\{0,\log x\}$, with $\log^+0=0$. For holomorphic $h$, $T$ is its Nevanlinna characteristic.

Statement recorded in the supplied source.

For arbitrary domains in this setup, without a simply-connectedness assumption, do \[I_\lambda(r,f)\le I_\lambda(r,f^*)\quad\text{for every }\lambda>0,\qquad T(r,f)\le T(r,f^*)\] hold for all $0<r<1$? [S2]

Short English statement

Does the universal cover of the circularly symmetrized image domain dominate every positive integral mean and the Nevanlinna characteristic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305016 (AMR-022-5016). 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 5.16, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.