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Problem

Let $\beta$ be a probability measure on $\mathbb{R}$, and define $$ K = \left \{z \in \mathbb{C}: g\left(z\right)=\int_{-\infty}^{\infty}\exp\left(z x\right)\beta ( dx ) \text{ is well-defined} \right\} $$

How do we do know if $g$ has any zeros in $K$? What are the conditions to ensure that there exists a strip $$ S= \left\{ z \in \mathbb{C}: \Re{z} \in I, \Im{z} \in \mathbb{R} \right\}$$ for some finite interval $I$ in $\mathbb{R}$ such that $g$ has no zeros in $S$?

Clearly the behavior of $g$ in $K$ depends critically on that of $\beta$ (including how $\beta$ distributes mass over $\mathbb{R}$ and the properties of its support). But I do not know how to tackle this problem generally. I would appreciate any of your suggestions.

Related References

(I edited this post as the previous version does not seems to be well phrased. Now the problem is in its general form, in that the measure probability $\beta$ can take many forms.)

A Google search using key words "zeros of laplace transform" returned two references by A. M. Sedletskii:

(a) one in "http://link.springer.com/article/10.1023%2FB%3AMATN.0000049682.65990.e7" identifies the curvelinear strip that contains all zeros of $g$ for compactly supported $\beta$ whose Radon-Nikodym derivative with respect to the Lebesgure measure $f$ is postive, nondecreasing and satisfies a local logarithmic convexity condition.

(b) the other in "www.tandfonline.com/doi/pdf/10.1080/10652469308819008 " that gives the necessary conditions for a compactly supported $\beta$ when the zeros of $g$, if existent, all lie in the left (or right) half plane.

These seem to be the only refrences on zeros of Laplace transforms for nontrivial cases. However, they have different targets than the propose question: to identify a strip in which $g$ has no zeros at all.

My own attempts

We know that $K$ is usually an infinite, vertical strip in $\mathbb{C}$ and $K = \mathbb{C}$ can hold. In the latter case, $g$ is entire and its zero set $E=\left\{z\in K: g(z)=0\right\}$ is at most countable. For this case, we know the Hadamard product. If $|E|=\infty$ then the sequence of zeros $\left\{ w_{n} \right\}$ converges to $\infty$ on the Riemman sphere. It seems intuitive that the strip $S$ free of the zeros of $g$ in this case should exist. For, if not, the Hadamard product may diverge (I guess), or, if any vertical strip, no matter how small the width is, contains at least one zero of $g$, then there exists a subsequence of $\left\{ w_{n} \right\}$ each of whose element has its real part arbitrarily close to a rational number in the interior of $P= \left \{x \in \mathbb{R}: x = \Re{z} \text{ for } z \in K \right\}$. However, I do not know how to proceed further.

On the other hand, if $K$ is a proper subset of $\mathbb{C}$, then by the Riemann mapping theorem, we can map $K$ into the unit disk $D=\left\{z\in\mathbb{C}: |z|<1\right\}$, and equivalently study the zeros of $h$ analytic in $D$. In this case, a vertical strip may be mapped into a fat spiral asymptotically approaching the boundary $\partial{D}=\left\{z \in \mathbb{C}:|z|=1\right\}$. For the case, I learnt the Blaschke product. However, I do not know if a Laplalce transform can display such behavior of Blaschke product as the properties of the Blaschke sequence are unclear to me. Additionally, I do not know if I should digger further into Nevanlinna theory for meromorphic functions in unit disk.

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