Let $f,g:G\to \mathbb{C}$ be holomorphic, and $\gamma$ a 1-cycle in $G$ which bounds an open set $A\subseteq G$. Suppose further that $|g(z)|<|f(z)|$ for all $z\in im(\gamma )$. Then the number of zeros of $f$ in $A$ is the same as 1;;the number of zeros of $f+g$ in $A$

PROOF:

• define homotopy between $f\circ {\gamma }_k$ and $(f+g)\circ {\gamma }_k$, with ${\gamma }_k$ being an element of the chain $\gamma$.