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

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?

• Maybe the discrete set $\mathbb Z$ is not enough, as there is the counterexample $F(x)=\exp(-x^2)\sin(x)$. Jan 3, 2022 at 11:40

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.