A note on Hall’s lemma

Abstract:

Let H be the half plane $\{ z:\operatorname {Re} z > 0\}$. Let y be a Jordan arc joining $z = 0$ and $z = {e^{i\alpha }}\;(0 \leqq \alpha < \pi /2)$ in $H \cap \{ |z| \leqq 1\}$. Let ${\gamma ^\ast }$ be the segment $z = iy\;(0 \leqq y \leqq 1)$ of the imaginary axis. If $\omega (z,\gamma )$ is the harmonic measure of $\gamma$ with respect to $H\backslash \gamma$ and $\omega (z,{\gamma ^\ast })$ the harmonic measure of ${\gamma ^\ast }$ with respect to H, then $\omega (x + iy,\gamma ) > \omega (x - i|y|,{\gamma ^\ast })$.