Unit disc ($\mathbb{D}$)

The unit disc in the complex plane is $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$.

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.