= 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 $g_K'(x)>0$. Use the course's Brownian excursion restriction identity: for an excursion from $x$ to infinity in the <complex upper half-plane>,
$$
\mathbb P_x^{\rm exc}(\widehat B\cap K=\varnothing)=g_K'(x).
$$
One way to obtain this identity is to start its <Doob h-transform> at $x+i\varepsilon$: its avoidance probability is $\operatorname{Im}g_K(x+i\varepsilon)/\varepsilon$, which tends to $g_K'(x)$ 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 $|x|>1$,
$$
\boxed{1-x^{-2}=g_D'(x)\leq g_K'(x)\leq1.}
$$
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.
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