The question is:
Map the domain between the two circles $(x−1)^2+y^2=1$ and $(x−2)^2+y^2=4$ conformally onto the upper half plane. Note that both circles are tangent to the y-axis at the origin.
With hint given: show that the mapping $w=\large \frac{1}{z}$ takes the region between the circles in the z plane to a strip in the w plane.
My attempt,
I tested 3 distinct points from the boundaries, which are the two circles, and under the mapping $\frac{1}{z}$, I see that the larger circle maps to a vertical line x = 1/4, and the smaller circle maps to a vertical line x = 1/2.
To line up this vertical strip with the y-axis, I subtracted off 1/4. Now my mapping is
$$ z \to f(z) = (\frac {1}{z} - \frac{1}{4}) $$
Now the image is a vertical strip with its left boundary line coinciding with the y-axis, and its right boundary line is x= (1/2 - 1/4), i.e., the line x = 1/4.
I then rotated this vertical strip of width 1/4 90 degrees, so that it becomes a horizontal strip of height 1/4, sitting on the real axis now.
$$ z \to f(z) = i(\frac {1}{z} - \frac{1}{4}) $$
I'm stuck with this strip in the upper half plane. What can I do now?
Thanks,