Schwarz's Lemma
Let
be a function on
with
and suppose there exists
such that
for
Then
for
maps the open discinto the the open disc
Proof: Sinceis analytic on we can write
as a Taylor series about 0.
for(
since)
Then
for
Thus the function
provides an analytic extension of
from 
to
Now apply the maximum principle to
on the open disc
where
(we cannot allow
since– and so– is not known to be continuous on
This gives
where
for
This inequality holds for all
such that
we deduce on letting
that
for
Hence
forso thatfor
but since this inequality obviously holds for z=0 we havefor
