Injective entire functions are affine ($g(z)=az+b,a\ne 0$)

An injective holomorphic function cannot have an essential singularity: Casorati-Weierstrass theorem would give values from a small punctured neighbourhood inside the open image of a disjoint neighbourhood, contradicting injectivity. Apply this at infinity to an injective entire function. A removable singularity there would make the function constant by Liouville's theorem. A pole makes it a polynomial by the Cauchy coefficient formula; its derivative has no zero, so the fundamental theorem of algebra forces degree one.