 Dominated perturbation preserves winding number

If closed continuous paths $\gamma ,\varphi$ satisfy $|\gamma (t)|>|\varphi (t)|$ for every $t$, then $H(s,t)=\gamma (t)+s\varphi (t)$ never vanishes. It is therefore a homotopy through closed paths in $\mathbb{C}\setminus \{0\}$, and