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].

中文翻译：

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由 Laplace-Stieltjes 变换定义的解析函数的结果，具有完美的 ϕ 类型

摘要 在本文中，我们引入了完美 ϕ \phi 型的概念，通过使用比通常更一般的函数来描述 Laplace-Stieltjes 变换最大分子的增长。基于这个概念，我们研究了由 Laplace-Stieltjes 变换定义的解析函数 F ( s ) F(s) 的逼近和增长在半平面上收敛，并获得了关于解析函数 F ( s ) 的充要条件的一些结果由 Laplace-Stieltjes 变换定义的 F(s) 具有完美的 ϕ \phi 类型，这是对 Kong 给出的先前结果的一些推广和改进 [关于 Laplace-Stieltjes 变换在右半平面解析的广义阶次和类型，数学学报。罪。59A (2016), 91–98], Singhal 和 Srivastava [关于由 Laplace-Stieltjes 变换表示的解析函数的近似，Anal。理论与应用 31 (2015), 407–420]。