Rouche's theorem from both of my resources say the following:
Let $C$ denote a simple closed contour and suppose that:
- Two functions $f(z)$ and $g(z)$ are analytic inside and on $C$
- $|f(z)|\gt |g(z)|$ at each point $\color{red}{\text{on } C}$.
Then $f(z)$ and $f(z)+g(z)$ have the same number of zeros, counting multiplicities.
Does it actually mean on, or does it mean on and within?
Mean $\color{red}{\rm on} \,C$. If $\gamma: [a,b]\to C$ is a parametrization, you can think that $|f(\gamma(t))|>|g(\gamma(t))|$, for all $t \in [a,b]$. But in practice we don't do this.
An example: $z^2+z$ has both zeros in $B(0,2)$. We have: $$|z| = 2 \implies |z| = 2 < 4 = |z^2|$$
So $z^2+z$ has the same quantity of zeros that $z^2$ in $B(0,2)$: two zeros.
Note that we don't need to worry about zeros on $C$: if $|f(z)|>|g(z)|$ on $C$, $f(z) = 0$ for some point in $C$ gives $0 > |g(z)|$, not good. And if $f(z)+g(z) = 0$, we get $f(z) = -g(z) \implies |f(z)| = |-g(z)| = |g(z)|$ and the inequality gives $|f(z)|>|f(z)|$, not good either.