= Sharp displacement bound for a compact H-hull
{title2=$|g_K(z)-z|\leq3r$}
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.
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