 Inverse-at-infinity criterion for local boundedness of a mapping-out function

The inverse of a hydrodynamically normalized mapping-out function has expansion $f(w)=w+O(1/w)$ at infinity. If bounded-domain points $z_n$ had unbounded images $w_n=g(z_n)$, then $z_n=f(w_n)=w_n+O(1/w_n)$ would be unbounded, a contradiction. Thus $g$ is bounded even near finite rough boundary points. Its own expansion at infinity then makes $g(z)-z$ uniformly bounded throughout the domain. Reflection is needed only near infinity, not along the entire hull boundary.