Research Problems in Function Theory — Problem 3.21

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 $\mathbb T=\partial\mathbb D$. A continuum is a nonempty compact connected set. For a continuum $K\subset\overline{\mathbb D}$, put $d=\operatorname{diam}K=\sup\{|z-w|:z,w\in K\}\in[0,2]$. The harmonic measure of $K$ seen from a point $z$ in the disc means the probability that planar Brownian motion started at $z$ first hits $K$ before exiting the disc, counting an exit through $K\cap\mathbb T$ as a hit. Equivalently, for $z\notin K$ it is the harmonic measure of $K$ on the boundary of the component of $\mathbb D\setminus K$ containing $z$. If $z\in K$, set this measure equal to $1$. Denote it by $\omega(z;\mathbb D;K)$.

Statement recorded in the supplied source.

Is it always true that \[\omega(0;\mathbb D;K)\ge\frac1\pi\arcsin\!\left(\frac d2\right)?\] Here $\arcsin$ takes values in $[0,\pi/2]$ on $[0,1]$. [S2]

Short English statement

Is the probability of hitting any continuum in the closed unit disc, starting at its centre, at least the stated function of the continuum’s diameter?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303021 (AMR-022-3021). 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. Problem 3.21, its complete Update 3.21, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.