The Schwarz Lemma is a relatively basic lemma in Complex Analysis, that can be said to be of greater importance that it seems. There is a whole article written on it.
The conditions and results of Schwarz Lemma are rather difficult to memorize offhand, some tips I gathered from the net on how to memorize the Schwarz Lemma are:
Conditions: holomorphic and fixes zero.
Result 1: can be remembered as “Range of f” subset of “Domain”.
can be remembered as some sort of “Contraction Mapping”.
Result 2: If , or , then where . Remember it as “ is a rotation”.
If you have other tips on how to remember or intuitively understand Schwarz Lemma, please let me know by posting in the comments below.
Finally, we proceed to prove the Schwarz Lemma.
Let be the open unit disk in the complex plane centered at the origin and let be a holomorphic map such that .
Then, for all and .
Moreover, if for some non-zero or , then for some with (i.e.\ is a rotation).
Since is analytic, on , and . Note that on , so is analytic on .
Let denote the closed disk of radius centered at the origin. The Maximum Modulus Principle implies that, for , given any , there exists on the boundary of such that
As we get , thus . Thus
Moreover, if for some non-zero or , then at some point of . By the Maximum Modulus Principle, where . Therefore, .