## 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 .

## Maximum modulus principle

Let be a function holomorphic on some connected open subset of the complex plane and taking complex values. If is a point in such that for all in a neighborhood of , then the function is constant on .

Informally, the modulus cannot exhibit a true local maximum that is properly within the domain of .