Harmonic Measure of Curves in the Disk
Donald E. Marshall and Carl Sundberg
A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constant $C \gt 0$ so that the harmonic measure of a subset $E$ of the closed unit disk $\overline{\Bbb D}$ evaluated at $0$ satisfies
(*) $\omega(0,E,{\Bbb D}\setminus E) \ge C \omega(0,E_{rad},{\Bbb D}),$
where $E_{rad}$ is the radial projection of $E$ onto $\partial{\Bbb D}$. FitzGerald, Rodin and Warschawski proved that if $E$ is a continuum in $\overline {\Bbb D}$ whose radial projection has length at most $\pi$ then (*) is true with $C=1$, and they asked how large the length, $|E_{rad}|$, can be in order for their result to be valid. We prove that (*) holds with $C=1$ provided $|E_{rad}|\le \theta_c \simeq 2\pi\left({{350}\over{360}}\right)$ and $\theta_c$ cannot be replaced by a larger number. Fuchs asked for the largest constant $C$ so that (*) holds for all $E$. We show that for every continuum $E\subset \overline{\Bbb D}$, (*) holds with $C=C_{2\pi}\simeq .977126698498665669\dots$, where $C_{2\pi}$ is the harmonic measure of the two long sides of a 3:1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) with $C=C_{2\pi}$.