Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $U$ be a bounded open subset in $\mathbf{H}$ contained in $\{\tau \in \mathbf{H}: \mathrm{Im}(\tau) > 1\}.$ Suppose that $f$ does not vanish on the closure of this open subset in $\mathbf{H}$.
Is the absolute value of $f$ bounded from below by a positive constant on $U$?
I'm thinking this should follow from applying the maximum-principle to $1/f$.