 Solution

At a real point $x$ outside the hull's closure, the Schwarz reflection principle makes the mapping-out function analytic across the boundary and $g_K^{\prime }(x)>0$. Use the course's Brownian excursion restriction identity: for an excursion from $x$ to infinity in the complex upper half-plane,One way to obtain this identity is to start its Doob h-transform at $x+i\epsilon$: its avoidance probability is $\mathrm{Im}{g}_K(x+i\epsilon )/\epsilon$, which tends to $g_K^{\prime }(x)$ by reflection. Thus the boundary derivative is an excursion avoidance probability, and in particular it is at most one.

Let $D$ be the filled unit half-disc. Since $K\subseteq D$, avoiding $D$ implies avoiding $K$. The monotonicity of boundary derivatives of mapping-out functions and part (b) give, for $|x|>1$,The upper bound is attained by the empty hull and the lower by the filled half-disc, so the comparison is sharp on either real tail.