Is a holomorphic function on the unit disc not vanishing at zero bounded

Let $f:\mathbf{D}\to \mathbf{C}$ be a holomorphic function on the unit disc. Suppose that $f(0) \neq 0$ and that $\vert f\vert$ is bounded from below by some real number $C>0$ on some annulus contained in $D$. Then, does it follow that $\vert f\vert$ is bounded from below on $\mathbf{D}$ by some positive real number $C^\prime$?

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Try $f(z)=1-z$. –  Did Oct 22 '11 at 15:08
Posted this as an answer to give you the opportunity to close this post. –  Did Oct 24 '11 at 9:23
@Did I do not understand the last line of the question and hence your example –  La Belle Noiseuse Apr 30 '13 at 14:23
@Tsotsi Hint: the answer to the question in the last sentence is "No". –  Did Apr 30 '13 at 16:37

Try $f(z)=1-z$.