A holomorphic function $f$, injective on $\partial D$, must be injective in $\bar{D}$? 
Prove: If $f$ is holomorphic on a neighborhood of the closed unit disc $\bar{D}$, and if $f$ is one-to-one on $\partial D$, then $f$ is
  one-to-one on $\bar{D}$.

(Greene and Krantz's Function Theory of One Complex Variable (3rd), Ch. 5, Problem 17.)
Can anyone provide a clue as to how to attack this problem ?

Some hints:
The function $f$ restricted to $\partial D$ is an injective continuous map of  $\partial D\sim S^1$ into ${\mathbb C}$. By the Jordan curve theorem the curve $\gamma:=f(\partial D)$ separates ${\mathbb C}\setminus\gamma$ into two connected domains $\Omega_{\rm int}$ and $\Omega_{\rm ext}$, called the interior and the exterior of $\gamma$. The points $a\in \Omega_{\rm int}$ are characterized by the fact that $\gamma$ has winding number $\pm1$ around them.
Create a connection with the argument principle.
