 
Summary: Geometry of the unit disc.
N.A.
13/2/07
Notation. C is the complex field; D = {z C : z < 1} is the unit disc;
S = D is the unit circle.
Theorem 1 (Schwarz' Lemma). Let f : D D be a holomorphic function and
suppose that f(0) = 0. Then
z D f(z) z and f (0) 1. (1)
Moreover, if equality holds in (1) for some z D or for the inequality involving
f (0), then v Sz D : f(z) = vz.
Proof. Let r (0, 1) and let gr(z) = f(rz)/z, gr : D C after removing the
singularity in z = 0. gr H(D) C(D) and
gr(ei
) = f(rei
) < 1 R,
hence, by the Maximum Principle1
, gr(z) < 1 for all z D, i.e., for any fixed
w D,
f(z)
z
