![]() Moreover, considerable attention has been paid to the growth and the value distribution of analytic functions defined by Dirichlet series and Laplace-Stieltjes transforms, and a great deal of interesting results focusing on the growth and value distribution of such functions can be found in (see ). In the past several decades, the problem on the growth and value distribution of analytic functions has been an important and interesting subject in the fields of complex analysis. Under some conditions related to, , and, the series ( 7) can converge in the whole plane or the half plane that is, is analytic in the whole plane or the half plane. Hence, we can say that Laplace-Stieltjes transform is a general form of Dirichlet series. For Dirichlet series ( 7), it can become a Taylor series if and, and it further can also become a classical Dirichlet series if, which is important in the fields of number theory. Where is real variables, is nonzero complex numbers. Then we can conclude from Theorem 1 that becomes a Dirichlet series Moreover, if is a step function, choosing a sequence such that , can become the classical Laplace integral form Moreover, it can be used in many fields of mathematics, such as functional analysis, and certain areas of theoretical and applied probability. Then ( 1) converges for every for which, andĪs we know, ( 1) can be called as Laplace-Stieltjes transform, which is an integral transform similar to the Laplace transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes. ![]() Where is a bounded variation on any finite interval, and and are real variables and obtained the following theorem. In 1946, Widder considered the convergence of the following form Our results exhibit the growth of Laplace-Stieltjes transforms from the point of view of approximation. The main aim of this paper is to establish some theorems concerning the error, the Sun’s type function, and of entire functions defined by Laplace-Stieltjes transforms with infinite order converge in the whole complex plane. ![]()
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