 Real boundary bounds for a unit-disc H-hull

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 $x<-1$.