A note on Hall’s lemma

Abstract

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