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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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I am sorry, I would like to understand the assumption: "$|f|$ is positive on a huge disc" means $|f|>0$ on $B(0,R)$ for some $R$ sufficiently large, is that right? –  Paul Dec 3 '11 at 2:13
    
Yes. Actually, you can assume f to have no zeroes on the whole of $\mathbf{C}$. –  Soka Dec 3 '11 at 7:35
    
No wait, assume it has no zeroes on, say $\mathbf{C} - \{0\}$. –  Soka Dec 3 '11 at 7:35
    
The setup is somewhat non-sensical because the only functions with these properties are $az^m$. What exactly was meant? –  fedja Oct 30 '12 at 19:57

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