Question. Is there an entire function $F$ satisfying first two or all three of the following assertions:
- $F(z)\neq 0$ for all $z\in \mathbb{C}$;
- $1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy space in the upper half-plane;
- $F$ is bounded in every horizontal half-plane $\{z\colon \text{Im}(z) > \delta\}$?
Thoughts. Let $G= 1/F$. Then we have $G(z) = 1 + \int_0^{\infty} h(x)e^{izx}\, dx$ for some $h\in L^2[0,\infty)$ and all $z\in \mathbb{C}_+$. For nice functions $h$ (e.g., for super-exponentially decreasing) this integral representation can be extended to the whole complex plane and probably the example can be constructed in terms of $h$. However, I don't know if it is possible to find $h$ such that $G$ is non-zero for every $z$.