 Solution

A compact H-hull is a bounded relatively closed subset $K$ of the complex upper half-plane such that $\mathbb{H}\setminus K$ is a simply connected domain. This does not require its closure to be connected. Its mapping-out function is the unique conformal map $g_K:\mathbb{H}\setminus K\to \mathbb{H}$ with hydrodynamic normalization at infinity,The half-plane capacity isThe coefficient is zero exactly for the empty hull.

In terms of the least radius of a real-centred enclosing half-disc, the sharp displacement bound for a compact H-hull, also called the continuity estimate, isThe differentiability estimate for a mapping-out function is a uniform small-hull expansion: there is an absolute constant $C$ such that if $K\subset \{z:|z-\xi |\le r\}$ with $\xi \in \mathbb{R}$, thenThese are statements of the two requested estimates. The second is not merely a bound for $g_K^{\prime }$: it controls the first-order change of a mapping-out function when a small hull is removed.