zbMATH — the first resource for mathematics

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

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)
Full Text: