Use Schwarz lemma, we can verify that $$f(z)=e^{i\theta}\frac{\alpha-z}{1-\overline{\alpha}z}$$exhaust all automorphisms of the disc($\theta \in \Bbb{R},\alpha \in \Bbb{D}$).
When I first see the problem: "Find all automorphisms of the disc." I don't know how to do to find the above analytic expression although it looks very simple. So how did people find this analytic expression in the past ? Why they knew this mapping $f(z)$ exhaust all the automorphisms of the disc?
This has been known for a very long time and it's probably difficult to trace the historical origin.
My guess is that Möbius transformations (or linear fractional transformations if you prefer) were very well understood at an early stage in the historical development of complex analysis. These mappings preserve the set of lines and circles, and it is natural to look for automorphism of the unit disc among the Möbius transformations that send the unit circle to itself.