On the Zeros of Laplace Transforms

A. M. Sedletskii

Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 883-892

Résumé

Suppose that $f$ is a positive, nondecreasing, and integrable function in the interval $(0,1)$. Then, by Pólya's theorem, all the zeros of the Laplace transform $$ F(z)=\int_0^1e^{zt}f(t)\,dt $$ lie in the left-hand half-plane $\operatorname{Re} z\le0$. In this paper, we assume that the additional condition of logarithmic convexity of $f$ in a left-hand neighborhood of the point $1$ is satisfied. We obtain the form of the left curvilinear half-plane and also, under the condition $f(+0)>0$, the form of the curvilinear strip containing all the zeros of $F(z)$.