Automorphism of the unit disk ($\varphi (z)=e^{i\theta }\frac{z-a}{1-\overline{a}z}$)

A biholomorphism of the unit disk to itself. Every such map has the form $e^{i\theta }(z-a)/(1-\overline{a}z)$ with $|a|<1$. These maps normalize an extremal univalent function in the proof of the Riemann mapping theorem.