Letbe a function onwithand suppose there existssuch thatfor
maps the open discinto the the open disc
Proof: Sinceis analytic on we can writeas a Taylor series about 0.
Thus the functionprovides an analytic extension offrom to
Now apply the maximum principle toon the open discwhere(we cannot allowsince– and so– is not known to be continuous on
This inequality holds for allsuch thatwe deduce on lettingthatfor
Henceforso thatforbut since this inequality obviously holds for z=0 we havefor