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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 $f(z)=1-z$.2011-10-22
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    Posted this as an answer to give you the opportunity to *close* this post.2011-10-24
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    @Did I do not understand the last line of the question and hence your example2013-04-30
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    @Tsotsi Hint: the answer to the question in the last sentence is "No".2013-04-30

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