Monotonicity of half-plane capacity

If compact H-hulls satisfy $A\subseteq C$, thenThis follows from the Brownian representation of half-plane capacity or from composition of their mapping-out functions.

If compact H-hulls satisfy $A\subseteq C$ and $D$ is the bounded filling of $g_A(C\setminus A)$, thenIn particular, inclusion is strict exactly when the capacity increase is positive.