The Growth of Entire Functions Defined by Laplace-Stieltjes Transforms Related to Proximate Order and Approximation

In this article, we discuss the growth of entire functions represented by Laplace-Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace-Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace-Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An*, and lambda n, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.