Rouché's Theorem

Theorem
Let $f$ and $g$ be complex-valued functions, holomorphic in the interior of some simply connected region $D$. If $\left\vert g\left({z}\right) \right\vert < \left\vert f\left({z}\right) \right\vert$ on the boundary of $D$, then $f$ and $f+g$ have the same number of zeroes in the interior of $D$ counted up to multiplicity.