On the Zeros of Laplace Transforms

Abstract

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

$$F(z) = \int_0^1 {e^{zt} f(t)dt} $$

lie in the left-hand half-plane $\operatorname{Re} z \leqslant 0$. 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 $f(z)$.