On the zeros of Laplace transforms. (English. Russian original) Zbl 1079.44001

Sedletskii, A. M.

Math. Notes 76, No. 6, 824-833 (2004); translation from Mat. Zametki 76, No. 6, 883-892 (2004).

The starting point is Polya’s theorem about the location of zeros of \[ F(z)=\int^1_0e^{zt}f(t)dt \] in the left half plane \(Re(z)\leq 0\), provided \(f\) is integrable, positive, and non-decreasing in \((0,1)\). First, the additional condition that \(f\) be logarithmically convex in a neighborhood of 1 is imposed and the form of the curvilinear half plane that contains the zeros is obtained. Next, if the condition that \(f(+0)>0\) is added, the left boundary of a strip containing the zeros can be determined. It is further shown that if \(f\) is logarithmically convex on \((0,1)\), not necessarily monotone, then the zeros are confined to the union of horizontal strips. Several corollaries also presented.

Reviewer: Robert G. Buschman (Bandon)

MSC:

44A10	Laplace transform
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Keywords:

Laplace transform; distribution of zeros; logarithmically convex function; regularly varying function