Unit-disc image from a boundary modulus lower bound

If $f$ is holomorphic near the closed unit disc, $|f|\ge 1$ on its boundary, and $|f(z_0)|<1$ at one interior point, then $f$ maps the open unit disc onto a set containing the open unit disc. Rouché's theorem first shows that $f$ has a zero and then that every $f-w$ with $|w|<1$ has one.