Uniqueness Theorems

A one-to-one conformal mapping which maps an open circular disc onto itself, with the center a fixed point, is a rotation.

Let two open annular regions each bounded by two concentric circles have the same outer boundary and be related by a one-to-one and conformal mapping. Also let the mapping be such that a circle of one region concentric with the boundary corresponds to a curve of the other region traversed in the same sense. Then the two annular regions are identical and the conformal mapping is a rotation.

There do not exist two distinct functions $w = f(z)$ which map the unit circle $\left|z\right| < 1$ in a schlicht manner onto a given region $\mathfrak {R}$, where the origin $z = 0$ is mapped onto a given point $w = w_{0}$ of $\mathfrak {R}$ and $f'(0) > 0$.

The only functions which map the unit circle $\left|z\right| < 1$ in a schlicht manner onto itself are the bilinear functions $e^{i\alpha } \dfrac {z-z_{0}}{1-\overline{z_{0}}z}$, $\alpha$ real, $\left|z_{0}\right| < 1$.