Suppose $a \in \mathbb{C}$, $|a| < 1$, and $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. How to prove dependence of $|f(z)|$ on $|z|$?
Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?