Can someone confirm whether or not my solution to the following question is okay? Or if I'm missing something
Question: Let f be an entire function satisfying: $|f(z)| \leq |\exp(z)| ,\: \forall z \in \mathbb{C}$. Show that there exist a $c \in \mathbb{C}$ such that $|c|\leq 1$ and $f(z) = c \cdot \exp(z), \: \forall z \in \mathbb{C}$.
Solution: Note that $\exp(z) ≠ 0, \: \forall z \in \mathbb{C}$. Hence the function $\dfrac{f(z)}{\exp(z)}$ is an entire function, with $\dfrac{|f(z)|}{|\exp(z)|} \leq 1$. Since it is bounded it must be constant and therefore $f(z) = c \exp(z)$ for some $c \in \mathbb{C}$. It is then seen that $\dfrac{|c \cdot \exp(z)|}{|\exp(z)|} \leq 1 \Leftrightarrow |c| \leq 1$
Thanks