We are looking for the lower bound on the \begin{align} |{\rm erf}(z)| \ge ??? \end{align} Here ${\rm erf}(z)$ is the complex-valued error function
We know that around $z=0$ the function is linear, and for $z \to \infty$, it is exponential. So we expect a bound of the form \begin{align} |{\rm erf}(z)| \ge C |z| e^{ - |z|^2} \end{align} However, I was not able to locate anything of this form in the literature.