An application of Rouché's theorem

Question

i need help to proof the next:

Let $f$ be analytic at $D$ minus a finite numbers of interior points where $f$ has poles. Show that if $0<|f(z)|<1$ over $\partial D$, then the number of poles of $f$ in $D$ is equal to the number of roots of the equation $f(z)=1$ in $D$.

I have tried to prove it using Rouché’s theorem, but i cannot conclude anything , I would appreciate your help.

Answer 1

HINT: Use the argument principle for $g(z)=1-f(z)$. What is the image of $\partial D$ under $g$?