Schwarz Lemma. Let $\backslash mathbb\; =\; \backslash$ be the open unit disk in the complex plane $\backslash mathbb$ centered at the origin and let $f\; :\; \backslash mathbb\backslash rightarrow\; \backslash mathbb$ be a holomorphic map such that $f(0)\; =\; 0$.
Then, $|f(z)|\; \backslash leq\; |z|\; \backslash \; \backslash forall\; z\; \backslash in\; \backslash mathbb$ and $|f\text{'}(0)|\; \backslash leq\; 1$.
Moreover, if $|f(z)|\; =\; |z|$ for some non-zero $z$ or $|f\text{'}(0)|\; =\; 1$, then $f(z)\; =\; az$ for some $a\; \backslash in\; \backslash mathbb$ with $|a|\; =\; 1$.〔Theorem 5.34 in 〕