In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk.
Schwarz’s Lemma: Let
be the open unit disk in the complex plane
. Let
be a holomorphic function with
. The Schwarz lemma states that under these circumstances
for all
, and
. Moreover, if the equality
holds for any
, or
then
is a rotation, that is,
with
.
This lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove; however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No similar result exists for real functions, of course. To prove the lemma, one applies the maximum modulus principle to the function
.
Proof: Let
. The function
is holomorphic in
(excluding
) since
and
is holomorphic. Let
be a closed disc within
with radius
. By the maximum modulus principle,

for all
in
and all
on the boundary of
. As
approaches
we get
. Moreover, if there exists a $z_0$ in
such that
. Then, applying the maximum modulus principle to
, we obtain that
is constant, hence
, where
is constant and
. This is also the case if
.
A variant of the Schwarz lemma can be stated that is invariant under analytic automorphisms on the unit disk, i.e. bijective holomorphic mappings of the unit disc to itself. This variant is known as the Schwarz-Pick theorem (after Georg Pick):
Schwarz-Pick theorem: Let
be holomorphic. Then, for all
,

and, for all 
.
The expression

is the distance of the points
in the Poincaré metric, i.e. the metric in the Poincaré disc model for hyperbolic geometry in dimension two. The Schwarz-Pick theorem then essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric. If equality holds throughout in one of the two inequalities above (which is equivalent to saying that the holomorphic map preserves the distance in the Poincaré metric) , then
must be an analytic automorphism of the unit disc, given by a Möbius transformation mapping the unit disc to itself.
An analogous statement on the upper half-plane
can be made as follows:
Let
be holomorphic. Then, for all
,
.
This is an easy consequence of the Schwarz-Pick theorem mentioned above: One just needs to remember that the Cayley transform

maps the upper half-plane
conformally onto the unit disc
. Then, the map
is a holomorphic map from
onto
. Using the Schwarz-Pick theorem on this map, and finally simplifying the results by using the formula for
, we get the desired result. Also, for all
,
.
If equality holds for either the one or the other expressions, then
must be a Möbius transformation with real coefficients. That is, if equality holds, then

with
,
,
,
being real numbers, and
.
Proof: The proof of the Schwarz-Pick theorem follows from Schwarz’s lemma and the fact that a Möbius transformation of the form
where 
maps the unit circle to itself. Fix
and define the Möbius transformations
and
.
Since
and the Möbius transformation is invertible, the composition
maps
to
and the unit disk is mapped into itself. Thus we can apply Schwarz’s lemma, which is to say
.
Now calling
(which will still be in the unit disk) yields the desired conclusion
.
To prove the second part of the theorem, we just let
tend to
.
Application 1 (QE Berkeley Spring 1991). Let the function
be analytic in the unit disc, with
and
. Assume that there is a number
such that
. Prove that
.
Solution. Schwartz’s lemma implies that the function
satisfies
. The linear fractional map
sends the unit disc onto itself. Applying Schwartz’s lemma to the function

we conclude that the function

satisfies
. Similarly, the map
sends the unit disc onto itself, and Schwartz’s lemma applied to the function

implies that the function
satisfies
. All together, then
.
which is the desired inequality.
Application 2 (QE NUS Spring 2009). Suppose
is analytic in
with
. Show that

for all
.
Application 3 (QE NUS Fall 2009). Is there an analytic function
on
(unit disk in the complex plane with center
) such that
for
with
and
? If so, find such an
. Is it unique?