Some New Inequalities on Laplace–Stieltjes Transforms Involving Logarithmic Growth

Abstract

This article is devoted to exploring the properties on the logarithmic growth of entire functions represented by Laplace–Stieltjes transforms of zero order. In order to describe the growth of Laplace–Stieltjes transforms more finely, we introduce some concepts of the logarithmic indexes of the maximum term and the center index of the maximum term of Laplace–Stieltjes transforms, and establish some new inequalities focusing on the above logarithmic indexes, the logarithmic order, the (lower) logarithmic type and the coefficients of Laplace–Stieltjes transforms. Moreover, we obtain two estimation forms on the (lower) logarithmic type of entire functions represented by Laplace–Stieltjes transform by applying these inequalities. One estimation is mainly by the center indexes of the maximum term, the other is by the logarithmic order, exponent and coefficients. Finally, we obtain the equivalence condition of entire functions with the perfectly logarithmic linear growth. This result shows that the two estimation forms can be equivalent to some extent.