I would like to show that the function
$$f=\frac{c-z}{1-\bar c z}$$
with $|c| < 1$ is holomorphic and an automorphism from the complex disk $\{z: |z|<1\}$ to itself? I'm looking for an elementary proof of this (e.g without using that we can characterize the rotations of the unit disk as in Automorphism of the unit disk that fixes a point)
The complex conjugate on the unit circle $z= e^{i \phi}$ is the inverse
$$\frac{e^{i \theta } r-e^{i \phi }}{1-r e^{i \phi -i \theta }}\cdot \frac{e^{-i \theta } r-e^{-i \phi }}{1-r e^{i \theta -i \phi }}=\frac{\left(e^{i \theta } r-e^{i \phi }\right) \left(e^{-i \theta } r-e^{-i \phi }\right)}{\left(1-r e^{i \phi -i \theta }\right) \left(1-r e^{i \theta -i \phi }\right)}=\frac{r^2 +1 -r e^{i (\theta -\phi )}-r e^{i (\phi -\theta )}}{r^2 +1 -r e^{i (\theta -\phi )} -r e^{i (\phi -\theta )}}$$
It follows that the unit circle is mapped onto itself. Since $z=0$ maps to $c,|c|<1$ the inner maps to the inner. Conformality is confirmed by the real 2d Jacobian, but that identical to independence of the map wrt $\overline z.$