Injective Entire Functions Are Linear

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function that is injective.

Prove that $f$ must be affine:

$$f(z) = az + b \qquad \text{for some } a,b\in\mathbb{C},\ a\neq 0.$$

Corollary to keep in mind: the biholomorphic automorphisms of $\mathbb{C}$ are exactly the maps $z\mapsto az+b$.

Hint (if you want one): study the behavior of $f$ near $\infty$, that is, the singularity of $g(w)=f(1/w)$ at $w=0$. Which of the three singularity types are compatible with injectivity?