Differentiability estimate for a mapping-out function ($|g_K(z)-z-\mathrm{hcap}(K)/(z-\xi )|\le Cr\mathrm{hcap}(K)/|z-\xi {|}^2$)

If a compact H-hull lies in a disc of radius $r$ centred at $\xi \in \mathbb{R}$, the displayed estimate holds for $|z-\xi |>2r$, with an absolute constant $C$. It refines the Laurent expansion uniformly in the hull and supplies the first-order increment needed to derive the Chordal Loewner equation from the Loewner local growth property.