Height contraction of a hydrodynamically normalized mapping-out function ($\mathrm{Im}{g}_K(z)\le \mathrm{Im}z$)

For a compact H-hull, $v(z)=\mathrm{Im}{g}_K(z)$ is harmonic and satisfies $0<v(z)\le \mathrm{Im}z$. 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 $v-\mathrm{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.