Analytic in an annulus

1. Jan 17, 2010

esisk

1. if f(z) is analytic on the unit disk D. Suppose there is an annulus U={r<|z|<1} such that the restriction of f(z) to U is one-to-one. Show that f(z) is one-to-one on D.

2. Relevant equations

I am having trouble using the hypotheseis "f is one-to-one" I believe. I cannot think other than this implying that f'(z) is nonzero, but this does not seem to suffice

3. The attempt at a solution
I am trying to use Swarz lemma in conjunction with Identity theorem (that is showing that two functions agree on an infinite set), but I