Real boundary bounds for a unit-disc H-hull ($x\le g_K(x)\le x+x^{-1}(x>1)$)

If a compact H-hull lies in the unit disc, compare its tail harmonic measure with that of the empty hull and filled half-disc. The Poisson kernel for the upper half-plane turns the inclusion of Brownian exit events into $x\le g_K(x)\le x+1/x$ for $x>1$. Reflection gives $x+1/x\le g_K(x)\le x$ for $x<-1$.

For a compact H-hull inside the disc of radius $r$ centred at a real point, its mapping-out function of a compact H-hull displaces every point of its domain by at most $3r$. After scaling and translation, the real boundary bounds for a unit-disc H-hull put the corresponding inverse boundary interval inside $[-2,2]$. Boundary cluster points of the inverse over this interval lie in the unit disc. The maximum modulus principle applied to $w-g_K^{-1}(w)$ gives the bound without assuming local connectedness of the hull. The nearly closed semicircular slit makes the constant three sharp.

With $m(z)=(z+1)/(1-z)$, the compact H-hull $K_L=m^{-1}(i(0,L])$ is a unit semicircular arc attached at $-1$, with a narrow passage near $1$. As $L\to \infty$, its normalized mapping-out function of a compact H-hull tends to $2$ at each fixed point of the interior bay. Points in that bay tending to $-1$ therefore have displacement approaching three. The order of these two limits is important.