Question: Suppose $f$ is analytic in a region A containing the unit disc $D = \{ z: |z| \leq 1 \} $ and such that $|f(z)|>2$ whenever $|z| = 1$. If $f(0) = 1$, show that $f$ has a zero in D.
Thoughts: I am studying for an exam in complex analysis and this is one of the more theoretical questions from an old exam. Since my knowledge of complex analysis theory is small, I'm not quite sure how to proceed. I supposed it has something to do with that $f$ must be negative somewhere on $D$ and since the function is analytic in the region, $f$ must have a zero somewhere according the mean-value theorem or an analogous version of it. All input is very much appreciated.