Research Problems in Function Theory — Problem 3.19

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.

A subharmonic function is an upper semicontinuous function with values in $[-\infty,\infty)$, not identically $-\infty$ on a connected component, that satisfies the mean-value inequality on every closed ball in its domain. A real harmonic function is a twice continuously differentiable function $u$ satisfying $\Delta u=0$. Let $\mathbb D=\{|z|<1\}$. A tangential path to $1$ is a continuous $\gamma:[0,1)\to\mathbb D$ with $\gamma(t)\to1$ and $(1-|\gamma(t)|)/|1-\gamma(t)|\to0$. Its rotation by $\theta$ is $\gamma_\theta(t)=e^{i\theta}\gamma(t)$. Absence of a limit means that the lower and upper limits differ, even allowing the value $+\infty$. The source update attributes an affirmative construction of an unbounded positive harmonic function to Aikawa.

Statement recorded in the supplied source.

For every such tangential path $\gamma$, is there a positive harmonic function $v$ on $\mathbb D$ such that \[\liminf_{t\to1}v(\gamma_\theta(t))<\limsup_{t\to1}v(\gamma_\theta(t))\qquad(0\leq\theta<2\pi)?\] [S2]

Short English statement

For any given tangential path, can one positive harmonic function fail to have a limit along every rotated copy of that path?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303019 (AMR-022-3019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 3.19 and Update 3.19. https://arxiv.org/abs/1809.07200.