Uniqueness results for holomorphic functions bounded in a strip with a certain decay to zero

Question

Let $A=\{ z \in \mathbb C : |\mathrm{Im}(z)| < a \}$ for some $a >0$. Further let $F:A \to \mathbb C$ be a holomorphic function on $A$ with the following properties:

1. $F$ is bounded on $A$
2. For every $y \in (-a,a)$ the map $x \mapsto |F(x + iy)|$ has a certain decay to zero as $|x| \to \infty$, for instance an exponential decay.

I'm wondering if for such functions certain uniqueness results holds true, in the sense that if $F$ vanishes on a certain discrete set on the real axis (say the integers $\mathbb Z$) then $F$ vanishes identically. Is anyone of you familiar with results in this direction?

Answer 1

Decrease on horizontal lines and density of zeros are two independent things.

1. A bounded function cannot have too many zeros. This is a consequence of Jensen's inequality which implies the Blaschke condition. The Blaschke condition is usually stated for the unit disk or for the upper half-plane. Use a conformal map of your strip onto the upper half-plane $z\mapsto\exp(\pi z/(2a))$. In your example, let $z_k=x_k+iy_k$ be the sequence of zeros; if $f\neq 0$ then the Blaschke condition becomes $$\exp\left(-\frac{\pi}{2a}|x_k|\right)\sin\left(\frac{\pi}{2a}|y_k|\right)<\infty.$$ If this series is divergent then $f=0$.

2. A bounded function cannot decrease too fast. This is a form of Phragmen-Lindelof theorem. In your situation, if $$\exp\left(\frac{\pi}{2a}|x|\right)\log|f(x+iy)|\to-\infty,$$ as $x\to+\infty$ or $x\to-\infty$, then $f=0$.

A good reference is is B. Ya. Levin, Lectures on entire functions, AMS, 1996. The mentioned form of the Phragmen-Lindelof theorem is Exercise 1 on p. 40, and Blaschke's condition in formula (6) on p. 219.