Schwarz lemma

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.

Schwarz Lemma. Let ${\displaystyle \mathbb {D} =\{z:|z|<1\}}$ be the open unit disk in the complex plane ${\displaystyle \mathbb {C} }$ centered at the origin and let ${\displaystyle f:\mathbb {D} \rightarrow \mathbb {D} }$ be a holomorphic map such that ${\displaystyle f(0)=0}$ and ${\displaystyle |f(z)|\leq 1}$ on ${\displaystyle \mathbb {D} }$.

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