# Variants of the Schwarz lemma

Take some self map on the unit disk $\mathbb{D}$, $f$. If $f(0) = 0$, $g(z) = f(z) / z$ has a removable singularity at $0$. On $|z| = r$, $|g(z)| \leq 1 / r$, and with the maximum principle on $r \to 1$, we derive $|f(z)| \leq |z|$ everywhere. In particular, if $|f(z)| = |z|$ anywhere, constancy by the maximum principle tells us that $f(z) = \lambda z$, where $|\lambda| = 1$. $g$ with the removable singularity removed has $g(0) = f'(0)$, so again, by the maximum principle, $|f'(0)| = 1$ means $g$ is a constant of modulus $1$. Moreover, if $f$ is not an automorphism, we cannot have $|f(z)| = |z|$ anywhere, so in that case, $|f'(0)| < 1$.