= 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>,
$$
g_K(z)=z+\frac{a_K}{z}+O(|z|^{-2}).
$$
The <half-plane capacity> is
$$
\boxed{\operatorname{hcap}(K)=a_K
=\lim_{z\to\infty}z\bigl(g_K(z)-z\bigr)\ge0.}
$$
The 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, is
$$
\boxed{|g_K(z)-z|\le3\operatorname{rad}(K),
\qquad z\in\mathbb H\setminus K.}
$$
The <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 $\xi\in\mathbb R$, then
$$
\boxed{\left|g_K(z)-z-\frac{\operatorname{hcap}(K)}{z-\xi}\right|
\le \frac{Cr\,\operatorname{hcap}(K)}{|z-\xi|^2},
\qquad |z-\xi|>2r.}
$$
These are statements of the two requested estimates. The second is not merely a bound for $g_K'$: it controls the first-order change of a <mapping-out function> when a small hull is removed.
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