Map out $A$ first. The image
$$
D=g_A(C\setminus A)
$$
with its bounded filling is a <compact H-hull>, and uniqueness of hydrodynamic normalization gives
$$
g_C=g_D\circ g_A.
$$
Comparing the coefficients of $1/z$ at infinity yields the <half-plane-capacity composition rule>
$$
\operatorname{hcap}(C)
=\operatorname{hcap}(A)+\operatorname{hcap}(D)
\geq\operatorname{hcap}(A).
$$
Thus \b[half-plane capacity is monotone under inclusion].

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 1 / b / ii / Solution

If compact H-hulls satisfy $A\subseteq C$ and $D$ is the bounded filling of $g_A(C\setminus A)$, then
$$
g_C=g_D\circ g_A,
\qquad
\operatorname{hcap}(C)
=\operatorname{hcap}(A)+\operatorname{hcap}(D).
$$
In particular, inclusion is strict exactly when the capacity increase is positive.


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 Incoming links: Half-plane-capacity composition rule