Solution
= Solution
Apply the maximum <modulus> principle to $f(z)/z$, with its removable value $f\prime(0)$, to obtain $|f(z)|\le|z|$. Equality at a nonzero interior point makes $f(z)/z$ constant, so $f(z)=az$ with $|a|=1$.
= Solution
Apply the maximum <modulus> principle to $f(z)/z$, with its removable value $f\prime(0)$, to obtain $|f(z)|\le|z|$. Equality at a nonzero interior point makes $f(z)/z$ constant, so $f(z)=az$ with $|a|=1$.