Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type

Abstract

In this paper, we introduce the concept of the perfect \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) 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) defined by Laplace-Stieltjes transforms with perfect \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].