There are some days that I am thinking in the following problem.
For any positive integer $x$, let $t(x)$ be a real number which a priori is such that $t(x)>1$ and $t(x)$ tends to $1$ as $x\to \infty$.
Now, denote $q=q(x)=e^{-2\pi t(x)}$ and $q_0=e^{-2\pi}$.
I have that $$ j(\tau(x))-j(i)=e^{2\pi t(x)}-e^{2\pi}+\sum_{k\geq 1}c_k(e^{-2k\pi t(x)}-e^{-2k\pi}). $$
Thus, I need an explicit lower bound for $j(\tau(x))-j(i)$ in terms of $t(x)-1$. I already used the Mean Value Theorem for reducing the problem to a lower bound for $$ j(\tau(x))-j(i)=2\pi (t(x)-1) \left(e^{2\pi \theta_0(x)}-\sum_{k\geq 1}kc_ke^{-2k\pi \theta_k(x)}\right), $$ for some $\theta_k(x)\in (1, t(x))$ (for all $k\geq 0$). Thus it suffices to find such an lower bound for the previous expression in parenthesis. However, I was not able to do it and, indeed, I am not sure that this is the best approach.
Any hint, reference or idea to follow?
Thanks in advance!
