Rouche's theorem: $g(z)$ has no roots in $L$, $|g(z)| > |f(z)|$ for the contour $\partial L$. Does $f(z)$ have no roots in $L$?

Question

Let $f(z)$ and $g(z)$ be analytic functions. Let $L$ be the complex unit disc and its contour is $\partial L$, the complex unit circle $|z| = 1$.

If $g(z)$ has no roots in $L$, e.g. $g(z) = z + 2$, and $|g(z)| > |f(z)|, z \in \partial L$, does that mean $f(z)$ has no roots in $L$?