= Solution
Write $D=\mathbb H\setminus K$. The <mapping-out function> is a <conformal map> $g_K:D\to\mathbb H$ with <hydrodynamic normalization at infinity>
$$
\boxed{g_K(z)=z+\frac{a_K}{z}+O(z^{-2})\qquad(z\to\infty).}
$$
Equivalently, $g_K(z)-z\to0$. The <Riemann mapping theorem> gives a map onto the half-plane. Since $K$ is bounded, infinity has an analytic real-boundary neighborhood. After sending its image to infinity, the <Schwarz reflection principle> gives a <Laurent series> $az+b+O(z^{-1})$ there, with $a>0$ and $b$ real. A real affine automorphism removes $a$ and $b$, giving the stated normalization. Reflection also makes $a_K$ real.
If $g_1,g_2$ both have this normalization, $g_2g_1^{-1}$ is a <conformal automorphism of the upper half-plane>. Such maps are real <Möbius transformations>. The expansion forces it to fix infinity, have leading coefficient $1$, and have zero translation. It is therefore the identity. Thus \b[the normalized <mapping-out function> exists and is unique].
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