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
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
maps the unit circle to itself. Fix and define the Möbius transformations
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?