= Solution
On $\mathbb H\setminus(A\cup C)$, the nonnegative <harmonic function> $h_A+h_C$ dominates the boundary data $\operatorname{Im}z$ on $A\cup C$: at a point of $A$, for example, $h_A(z)=\operatorname{Im}z$ and $h_C(z)\geq0$. The <maximum principle for harmonic functions>, or equivalently <Brownian motion> stopped on the union, gives
$$
h_{A\cup C}(z)\leq h_A(z)+h_C(z).
$$
Comparing the coefficients at infinity proves
$$
\boxed{\operatorname{hcap}(A\cup C)
\leq\operatorname{hcap}(A)+\operatorname{hcap}(C)}.
$$
Back to article page
