Schwarz's Lemma

Letbe a function onwithand suppose there existssuch thatfor

Then for

maps the open discinto the the open disc

Proof: Sinceis analytic on we can writeas a Taylor series about 0.

for(since)

Thenfor

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 gives

where

for

This inequality holds for allsuch thatwe deduce on lettingthatfor

Henceforso thatforbut since this inequality obviously holds for z=0 we havefor