I need to find a map that takes the region between two circles $|z|=1$ and $|z-1/4| = 1/4$ to an annulus $a<|z|<1$.
Now I know that the bilinear transform $f(z) = \frac{z-\alpha}{1-\bar{\alpha}z}$ maps the unit disk to itself, so I've constructed 2 maps, one that takes the unit disk to itself, and another that takes the inside of the smaller circle to the disk $|z| < a$:
$f_1(z) = \frac{z-\alpha_1}{1-\bar{\alpha}_1z}$
$f_2(z) = a\big(\frac{4(z-1/4)-\alpha_2}{1-\bar{\alpha}_2(z-1/4)}\big)$
The idea is that if I can find a map that simultaneously does these two things, I'll have my answer. Unfortunately, I'm not sure how to 'combine' these maps. Obviously composition isn't the answer since I want simultaneous mapping, not sequential mappings.
Can someone help me equate these?