Closed Ball center subset upper Half Plane

If $F$ is a modular form where $F(it)$ is positive for sufficiently large $t$ (i.e. constant term is positive) and the derivative is positive, then $F$ is also positive.

Let $F : \mathbb{H} \to \mathbb{C}$ be a holomorphic function where $F(it)$ is real for all $t > 0$. Assume that Serre derivative $\partial_k F$ is positive on the imaginary axis. If $F(it)$ is positive for sufficiently large $t$, then $F(it)$ is positive for all $t > 0$.

For a cusp form f of level Γ(n), we have t^s * f(it) → 0 as t → ∞ for any real power s. This follows from the exponential decay of cusp forms at infinity: f = O(exp(-2π τ.im / n)).