On the Zeros of Laplace Transforms

摘要

Suppose that f is a positive, nondecreasing, and integrable function in the interval (0, 1). Then, by Polya's theorem, all the zeros of the Laplace transform F(z) = ∫ from x=0 to x=1 of e~(zt)f(t)dt lie in the left-hand half-plane Re z ≤ 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).