= Height contraction of a hydrodynamically normalized mapping-out function
{title2=$\operatorname{Im}g_K(z)\leq\operatorname{Im}z$}
For a <compact H-hull>, $v(z)=\operatorname{Im}g_K(z)$ is harmonic and satisfies $0<v(z)\leq\operatorname{Im}z$. Local boundedness and the interior continuity of $g_K^{-1}$ show that $v$ tends to zero at every finite boundary point. The <maximum principle for harmonic functions> applied to $v-\operatorname{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.
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