I need some help with the following problem from Ahlfors' Complex Analysis.
Problem: Find a single Möbius transformation $\phi$ (that is, a map of the form $\phi(z) = \dfrac{az + b}{cz + d}$, where $a,b,c,d$ are complex numbers) that maps the circles $|z| = 1$ and $\left|z - \frac{1}{4}\right| = \frac{1}{4}$ to concentric circles. Infinite values are fair game; I'm working on the Riemann sphere $\mathbb{C}_{\infty}$.
What I tried: I know that the family of maps $z \mapsto \dfrac{z - a}{1 - \bar{a}z}$, where $a \in \{z : |z| < 1 \}$, preserves the unit disk, and therefore also the circle $|z| = 1$. I then used the following facts:
(i) $\left|z - \frac{1}{4}\right| = \frac{1}{4}$ should be mapped to some circle inside the unit disk centered at $0$. Call this circle $C$.
(ii) $0$ is in the border of $\left|z - \frac{1}{4}\right| \leq \frac{1}{4}$, so $\phi(0) = a$ should go to the border of $C$. This means $C$ has radius $|\phi(0)| = |a|$. By rotating $C$, we can assume $a$ is positive and real.
(iii) $1/2$ is also in the border of $\left|z - \frac{1}{4}\right| \leq \frac{1}{4}$, and therefore it should also be mapped to the border of $C$. By solving the equation
$\left|\phi(1/2)\right| = |a|$
for $a$, where $a$ is real and positive because of the above reasoning, I should get the desired $a$, but instead I get functions that don't map $|z - 1/4| = 1/4$ to some circle centered at $0$. And now I'm stuck.
Any help appreciated!