= Injective entire functions are affine
{title2=$g(z)=az+b,\quad a\ne0$}
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.
Back to article page
