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.