Research Problems in Function Theory — Problem 3.1

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 $\Delta u=0$. A harmonic polynomial on $\mathbb R^2\cong\mathbb C$ is a polynomial in the two real coordinates that is harmonic. A path to infinity is a continuous map $\gamma:[0,\infty)\to\mathbb C$ with $|\gamma(t)|\to\infty$. The source update gives counterexamples of Barth, Brannan and Hayman even for growth powers $\alpha\geq1/2$.

Statement recorded in the supplied source.

For every harmonic function $u$ on the plane that is not a polynomial and every integer $n\geq1$, does there exist a path $\gamma_n$ to infinity with \[\frac{u(\gamma_n(t))}{|\gamma_n(t)|^n}\longrightarrow+\infty?\] Can one choose a single path $\gamma$ for which this holds for every integer $n\geq1$? [S2]

Short English statement

Must a nonpolynomial harmonic function grow faster than any prescribed power along some path, and can one path work for all powers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303001 (AMR-022-3001). 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 3.1 and Update 3.1. https://arxiv.org/abs/1809.07200.