Möbius Transformations <=> holomorphic and 1-to-1?

[nonequilibrium]

So every Möbius transformation of the complex plane is holomorphic and 1-to-1 on the Riemann sphere. Is the converse also true, or are there counter-examples?

 

[mfb]

I think so.

The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold

https://en.wikipedia.org/wiki/Möbius_transformation