Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 3 / b

Set $h_A(z)=\mathrm{Im}(z-g_A(z))$. This is a nonnegative harmonic function on $\mathbb{H}\setminus A$, has boundary value $\mathrm{Im}z$ on the hull boundary, and tends to zero on the real boundary. If $\tau$ is the first exit time of planar Brownian motion from $\mathbb{H}\setminus A$, the optional stopping theorem givesThe hydrodynamic expansion yields $h_A(iy)=\mathrm{hcap}(A)/y+O(y^{-2})$, and therefore the Brownian representation of half-plane capacity

If $A\subseteq C$, Brownian motion exits $\mathbb{H}\setminus C$ no later than it exits $\mathbb{H}\setminus A$. The Strong Markov property and the nonnegative harmonic function $h_A$ show that the expected exit height for $C$ is at least that for $A$. Taking the limits in the Brownian representation of half-plane capacity proves the monotonicity of half-plane capacity

On $\mathbb{H}\setminus (A\cup C)$, the nonnegative harmonic function $h_A+h_C$ dominates the boundary data $\mathrm{Im}z$ on $A\cup C$: at a point of $A$, for example, $h_A(z)=\mathrm{Im}z$ and $h_C(z)\ge 0$. The maximum principle for harmonic functions, or equivalently Brownian motion stopped on the union, givesComparing the coefficients at infinity proves