Let be a Mobius trasformation, so with . There are two cases to consider. One case is easy, the other case is more difficult. The first case is when , when this happens we just have . Thus, is just a translation, rotation, and magnification. Under these conditions the only way to map a unit circle to a unit circle is when the magnification factor is 1, and the translation is zero. However, you can rotate the circle as much as you like, therefore .

In the second case the Mobius transformation will always have a singularity (at ). The singularity cannot lie on the circle because otherwise the circle will get mapped into a line. The singularity either lies inside the circle or outside the circle. If it lies outside the circle then the maps the inside of the disk onto itself and the boundary of the disk onto itself by the maximum modolus theorem. It can now be shown that must have the form where and conversely. So this is the complete description of Mobius transformations which have their singularity outside the circle. I did not solve the problem yet of what happens if the singularity lies inside of the circle, I am thinking of considering the function but so far I was unable to complete the argument in the second case.