Rouche’s theorem

Let

1. $D$ - bounded and simply-connected
2. $f(z)$ and $g(z)$ are regular in $D \cup \delta D = \overline{D}$
3. $|g(z)| < |f(z)|,, \forall z \in \delta D$

Then functions $f(z)$ and $f(z) + g(z)$ has the same number of zeros in $D$.