Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{x+iy: 0\leq x <1 , y> \frac{1}{\epsilon}\}\subset \mathbf{H}$$ under the quotient map $\mathbf{H}\longrightarrow X(1)$. Note that $B(\epsilon)$ is an open disc around the cusp $\infty$.
Is the absolute value of the $j$-invariant bounded from below (by a positive real number depending on $B(\epsilon)$ in terms of \epsilon$) on $\epsilon$?B(\epsilon)$?
The answer to this is probably no. But what if we take an annulus?
Is the absolute value of the $j$-invariant bounded from below on $B(\epsilon) - B(\epsilon/2)$ ?in terms of $\epsilon$?
