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Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $q:\tau\mapsto \exp(\pi i \tau)$ be the nome on $\mathbf{H}$. Suppose that there are integers $a_j$ such that $$f(\tau)= \sum_{j=1}^\infty a_j q(\tau)^j$$ on $\mathbf{H}$.

In my case, the coefficients $a_j$ are positive for $j$ odd and negative for $j$ even. Moreover, I don't have explicit formulas for the $a_j$, but I have explicit bounds on $\vert a_j\vert$. In fact, $\vert a_j\vert \leq j 10^j$.

Now, I restrict my function $f$ to the open subset $$U=\{x+iy : -\frac{1}{2} < x < \frac{1}{2}, \frac{1}{2} <y < 2\}\subset \mathbf{H}.$$ Then, the absolute value of $f|_{U}$ takes its minimum at the boundary of $U$. Moreover, this minimum is known to be a positive real number.

Is it possible to write down an explicit (non-trivial) lower bound for $\vert f|_{U}\vert$ using just the above information?

The application I have in mind is to certain $q$-expansions of modular forms.

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I don't see what is preventing a sequence of such functions from having zeroes that quickly approach the boundary from outside. You can turn each series into a formal infinite product, and fine-tune the radius of convergence. –  S. Carnahan Oct 24 '11 at 12:44
    
Yeah. But in my case the function $f$ has no zeroes on $\mathbf{H}$. It vanishes at $i\infty$ and resembles in this way a bit the modular lambda function. –  Alfonso Oct 24 '11 at 15:18
    
Oh, you said "a sequence" of such functions. Why is that relevant? –  Alfonso Oct 24 '11 at 15:20
    
Is $a_1 = 1$, then (which is usually the case in constructions of modular units)? –  Noam D. Elkies Oct 24 '11 at 18:49
    
I think you need better bounds on $a_j$ to even be sure the series will converge on $U$. For instance, if $\tau\sim i/2$, then $q\sim\exp(-\pi/2)\geq 1/5$, so you would at least need $a_j=O(5^j)$. –  Kevin Ventullo Oct 25 '11 at 3:00
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