Abstract
Abstract In this paper, we introduce the concept of the perfect ϕ \phi -type to describe the growth of the maximal molecule of Laplace-Stieltjes transform by using the more general function than the usual. Based on this concept, we investigate the approximation and growth of analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms convergent in the half plane and obtain some results about the necessary and sufficient conditions on analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms with perfect ϕ \phi -type, which are some generalizations and improvements of the previous results given by Kong [On generalized orders and types of Laplace-Stieltjes transforms analytic in the right half-plane, Acta Math. Sin. 59A (2016), 91–98], Singhal and Srivastava [On the approximation of an analytic function represented by Laplace-Stieltjes transformations, Anal. Theory and Appl. 31 (2015), 407–420].
Highlights
Let L(s, F, α) be a class of Laplace-Stieltjes transforms +∞∫ F(s) = esxdα(x), s = σ + it, (1.1)where α(x) is a bounded variation on any finite interval [0, Y](0 < Y < +∞), and σ and t are real variables
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( ) if F(s) ∈ L0 is of finite order ρ, let φ(σ) =
Summary
[16] If Laplace-Stieltjes transform F(s) ∈ L0, and is of order ρ (0 < ρ < ∞) and of type T, for any real number −∞ < β < 0, we have ρ lim sup n→+∞ If φ ∈ Ξ0 and Laplace-Stieltjes transform F(s) ∈ L0 satisfies lim sup σ→0− If Laplace-Stieltjes transform F(s) ∈ L0 satisfies lim sup −log(−σ) = 0, σ→0− If Laplace-Stieltjes transform F(s) ∈ L0 satisfies (1.7), lim log+Mu(σ, F) σ→0− φ(σ) lim sup n→+∞
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