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 $C^\prime$?
Is a holomorphic function on the unit disc not vanishing at zero bounded
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complex-analysis
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6Try $f(z)=1-z$. – 2011-10-22
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0Posted this as an answer to give you the opportunity to *close* this post. – 2011-10-24
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0@Did I do not understand the last line of the question and hence your example – 2013-04-30
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0@Tsotsi Hint: the answer to the question in the last sentence is "No". – 2013-04-30