Let 
be analytic on the unit disk, and assume that
1. 
for all 
,
and
2. 
for some 
,
the unit disk.
Then
 |
(1)
|
Furthermore, if 
and 
,
then
 |
(2)
|
where 
is the complex conjugate (Krantz 1999, p. 78).
As a consequence, if either
 |
(3)
|
or
 |
(4)
|
for 
,
then 
is a conformal self-map of 
to itself.
Stated succinctly, the Schwarz-Pick lemma guarantees that if 
is an analytic map of the disk 
into 
and 
preserves the hyperbolic distance between any two points,
then 
is a disk map and preserves all distances.
