Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 203 / 2 / c

Compare the hull with the empty hull and the filled unit half-disc $J=\{z\in \mathbb{H}:|z|\le 1\}$. Their mapping-out functions of compact H-hulls are $g_{\varnothing }(z)=z$ and $g_J(z)=z+1/z$, the latter also giving the half-plane capacity of a half-disc.

Use one planar Brownian motion started at $iy$, with $y>1$. For $x>1$, exiting $\mathbb{H}\setminus J$ through $(x,\infty )$ entails avoiding $K$, while exiting $\mathbb{H}\setminus K$ through this interval entails reaching it before exit from the whole half-plane. Thus the tail harmonic measures satisfyFor any of these fixed hulls $L$, the Poisson kernel for the upper half-plane gives the exact tail formulaSubtract from $1/2$, multiply by $\pi y$, and take the limit. The inequalities reverse, yielding the real boundary bounds:Applying this result to the reflected hull $-\overline{K}$ also givesThe comparison is of Brownian exit events, so it does not assume that arbitrary conformal maps are pointwise ordered by domain inclusion.