Suppose $f$ is entire and $|f(z)|\geqslant |z|$ $\forall z \in \mathbb{C}$, prove there exists $c \in \mathbb{C} $ such that $f(z) = cz $ $\forall z \in \mathbb{C}$.
I want to use Liouville’s theorem for $\frac{z}{f(z)}$ but I don’t know what to do when $z = 0$ and $f(z) \neq 0$.
The only possible zero of $f$ is at the origin, so that $g(z) = z/f(z)$ is holomorphic in $\Bbb C\setminus \{ 0 \}$.
$g$ bounded ($|g(z)| \le 1$) and therefore has a removable singularity at $z=0$, i.e. it can be extended to a holomorphic function on $\Bbb C$ (Riemann's theorem on removable singularities).
The extended function is bounded and therefore constant (Liouville's theorem).