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Let $f$ be a holomorphic function on $\mathbf{C}$ and consider its restriction to the annulus $X=B(0,1) - B(0,3/4)$ in the complex plane. (Here $B(0,r)$ is the open disc with radius $r$ around $0$.) Assume $\vert f\vert$ is positive on a huge disc.

Let's suppose that $f=\sum a_j z^j$ on $\mathbf{C}$, where $a_j$ is an integer. Suppose that $\vert a_j \vert \leq 3^j$. Can we give a lower bound for $\vert f\vert$ on $X$ using that $3/4 < \vert z\vert < 1$?

If not, under what extra conditions would this be possible?

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    The setup is somewhat non-sensical because the only functions with these properties are $az^m$. What exactly was meant?2012-10-30

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