Automorphisms of the Riemann sphere

A holomorphic self-map of the sphere is a rational function. If it is bijective, its topological and algebraic degree is $1$, so it is the quotient of two linear polynomials with nonzero determinant. Alternatively, use sharp three-transitivity to compose the automorphism with a Möbius map fixing $0,1,\infty$, then show the resulting automorphism is the identity.

Holomorphic self-maps of the sphere need not be automorphisms: every nonconstant rational function gives such a map, and maps of degree greater than $1$ are branched coverings rather than bijections. Orientation-reversing conformal automorphisms are anti-Möbius, not holomorphic.

1. Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§8–9.