F₂ tendsto at Im Infty

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 $t > 0$.

Closed ball centered at z with radius z.im/2 is contained in the upper half plane.