Let $C$ be a simple closed curve, and let $f(z)$ be analytic inside and on $C$, $f(z) \neq 0$ on $C$. Then how can I prove the followings?
If $f(z)$ has no zeros inside $C$, then $f(C)$ does not surround the origin.
If $f(z)$ does have zeros inside $C$, then $f(C)$ must wrap around the origin.