Complex mapping of an annulus problem

Question

I am having a hard time understanding how to set up this map.

$f$ maps the annulus holomorphically onto itself but permutes the inner and outer boundaries $|z| = a$ and $|z| = b$, $0 < a < b$. Find the expression for $f$.

I have tried to rescale a to 1 among other things, but for some reason, I fail to see how can I set up this map.

Answer 1

Hint: Let $w = \dfrac{ab}{z}$.