Bounding a q-expansion on a bounded open subset of the complex upper-half plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:38:07Z http://mathoverflow.net/feeds/question/78966 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78966/bounding-a-q-expansion-on-a-bounded-open-subset-of-the-complex-upper-half-plane Bounding a q-expansion on a bounded open subset of the complex upper-half plane Alfonso 2011-10-24T11:25:19Z 2011-10-24T11:25:19Z <p>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 $\mathbf{H}$. </p> <p>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, $\vert a_j\vert \leq j 10^j$.</p> <p>Now, I restrict my function $f$ to the open subset <code>$$U=\{x+iy : -\frac{1}{2} &lt; x &lt; \frac{1}{2}, \frac{1}{2} &lt;y &lt; 2\}\subset \mathbf{H}.$$</code> 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.</p> <p>Is it possible to write down an explicit (non-trivial) lower bound for $\vert f|_{U}\vert$ using just the above information?</p> <p>The application I have in mind is to certain $q$-expansions of modular forms.</p>