This problem is from Section VIII.6 of Theodore Gamelin's Complex Analysis:
Let $f(z)$ be analytic on an open set containing a closed path $\gamma$, and suppose $f(z)\neq0$ on $\gamma$. Show that the increase in arg$f(z)$ around $\gamma$ is $2\pi W(f\circ\gamma,\, 0)$.
Any hints?