Rouche's theorem: Suppose $f,g$ are analytic inside and on a regular closed curve $\gamma$ and that $|f| > |g|$ for all $z\in \gamma$, then $\mathbb{Z}(f+g)=\mathbb{Z}(f)$ inside $\gamma$, where $\mathbb{Z}(f)$ is the number of zeros of $f$.
My question is: What about the zeros of $f$ and $g+f$ on $\gamma$?