Liouville and L'Hopital

Question

Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function such that $|f(z)|\geq |e^z-1|$ for all $z\in \mathbb{C}$. Prove there exists a constant $A$ such that $f(z)=A(e^z-1)$. My attempt: define $h(z)=\frac{f(z)}{e^z-1}$. It can be easily shown that if $f(z)=0$ then $z=2\pi ik$ for $k\in\mathbb{Z}$. I want to show we can analytically extend $h$ to the points $2\pi ik$ but the problem is that I can't use L'Hopital's rule because I don't know if $f(2\pi ik)=0$. How to progress from here?