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?

The problems from NUS QE are from GTM by Conway.

Comment by roticv — September 23, 2010 @ 14:53

Hi, thanks so much, btw, which problem from NUS you are mentioning? I do think the first one is standard but the second one may be new.

Comment by Ngô Quốc Anh — September 23, 2010 @ 16:16

Both questions and it’s given in the same order as in the book. The book has a hint to Spring 2009.

Comment by roticv — September 27, 2010 @ 1:01

Thanks. I will check it.

Comment by Ngô Quốc Anh — September 27, 2010 @ 1:03

Thanks roticv, I got it. They are on pages 132 and 133 respectively of the first book.

Comment by Ngô Quốc Anh — September 27, 2010 @ 1:07

can anybody please send me the answer for the “Application 2 (QE NUS Spring 2009) and Application 3 (QE NUS Fall 2009)” displayed above.

Thanks in advance.

Comment by ccc — March 7, 2012 @ 16:55

Go read the Conway book for details.

Comment by Ngô Quốc Anh — March 7, 2012 @ 18:03

yeah,I found the question in Conway.But I can’t find a clear method to solve the application 2 above.What will happen if such a function exist?how can we find that function?

Comment by ccc — March 8, 2012 @ 8:07