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On proximate order and type function of Laplace-Stieltjes transformations convergent in the right half-plane. (English) Zbl 1269.30033

The authors introduce the proximate order and type-function for analytic functions of finite order represented by Laplace-Stieltjes transformations \(F(s)=\int_{0}^{+\infty}e^{-sy}d\alpha(y), \;s=\sigma+it\), which is convergent only in the right half-plane, give two necessary and sufficient conditions and investigate the growth of such functions.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
44A10 Laplace transform
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