First we recall the statement of Schwarz’s lemma, which is a basic result in complex analysis. Let be the unit disk in .
Schwarz’s lemma. Let . If for all and , then , and for all . If or for some , then for some with .
To prove this, consider the holomorphic function and apply the maximum principle. (See any textbook for details.) Note that is maximized if and only if is a conformal self-map of (with the normalization ). If we do not specify the value of , we get Pick’s lemma.
Pick’s lemma. Suppose is holomorphic. Then
for any . Equality holds for some if and only if is a conformal self-map of (and in that case equality holds everywhere).
Pick’s lemma leads naturally to the hyperbolic metric on .
Schwarz’s lemma has the following useful Corollary called the Principle of Subordination. It makes clear the role of conformal maps in Schwarz’s lemma. First we need a notation. If is holomorphic (or merely continuous) on and , we define
Principle of Subordination. Let . Suppose that , is one-to-one, and , then and for all .
Proof. Since is one-to-one, is in fact a conformal map from to . Let be the inverse of . Then is holomorphic and maps into with since . By Schwarz’s lemma, we have
for all . By the chain rule, . So the first inequality gives .
Fix . The second inequality implies that
It follows that
Note that the second last equality follows from the maximum principle.
As an exercise, formulate the condition for equality. Here we give a simple example.
Example. Let be the right half plane. Let be holomorphic. We claim that .
Proof. Write , where . Define . Then is a conformal map from to with . Clearly . By the principle of subordination, we have . By direct calculation, and the result follows.
Using this estimate, we can prove the following interesting result.
Example (a prelim problem). Let be holomorphic. Then .
Proof. Since omits , there exists such that . Note that
Since , we have . It follows that maps into . Let . By the above result, we have
It follows that
Maximizing the last expression over , we have
More examples will be added later.