The Singular Points of Analytic Functions with Finite X-Order Defined by Laplace-Stieltjes Transformations

We study the singular points of analytic functions defined by Laplace-Stieltjes transformations which converge on the right half plane, by introducing the concept of X-order functions. We also confirm the existence of the finite X-order Borel points of such functions and obtained the extension of the finite X-order Borel point of two analytic functions defined by two Laplace-Stieltjes transformations convergent on the right half plane. The main results of this paper are improvement of some theorems given by Shang and Gao.


Introduction
For Laplace-Stieltjes transforms where () is a bounded variation on any interval [0, ] (0 <  < +∞) and  and  are real variables.We choose a sequence which satisfies the following conditions: Remark 1. Dirichlet series was regarded as a special example of Laplace-Stieltjes transformations; a number of articles have focused on the growth and the value distribution of analytic functions defined by Dirichlet series; see [1][2][3] for some recent results.
In 1963, Yu [4] proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of Laplace-Stieltjes.
Theorem A. Suppose that Laplace-Stieltjes transformations (1) satisfy the first formula of (3) where    is called the abscissa of uniform convergence of ().
In the past few decades, many people studied some problems of analytic functions defined by Laplace-Stieltjes transformations and obtained a number of interesting and important results (including [5][6][7][8][9][10][11]).In those papers, there are about two methods to control the growth of the maximum modulus   (, ) or the maximum term (, ): one method is to replace the denominator in the definition of growth order by using the technique of type function () (see [4,[12][13][14]) and the other method is to take multiple logarithm to   (, ) or (, ) in the definition of growth order (see [15,16]).For the second method, as the logarithm function is a special function, a question rises naturally: whether we can find a relatively general function to replace the logarithm function to control the maximum growth rate.
In this paper, we investigate the above question and give a positive answer to this question.Moreover, we confirm the existence of singular points for these functions by applying the main results of our paper.To do this, we introduce a completely new technique based on the concept of () which is different with the type function () and more general than logarithm function and obtain the main theorems as follows.
Remark 5.The definitions of -order and the function in Theorems 3 and 4 () will be introduced in Section 2.
From Theorem 4, we further investigate the value distribution of analytic functions with finite -order represented by Laplace-Stieltjes transformations convergent on the right half plane and obtain the following theorems.The structure of this paper is as follows.In Section 2, we introduce the concept of -order and give the proofs of Theorems 3 and 4. Section 3 is devoted to proving Theorems 6 and 7.
Remark 8. From Theorems 3-7, we assume that the -order  * of () is finite; that is, 0 <  < ∞.For  * = +∞, we have studied the value distribution of analytic functions defined by Laplace-Stieltjes transformations which converge on the right half plane and obtained some results (see [17]).

Proofs of Theorems 3 and 4
We first introduce the concept of -order of such functions as follows.
To control the growth of the molecule   (, ) or (, ) in the definition of order, many mathematicians proposed the type functions () to enlarge the growth of the denominator log(1/) or − (see [4,6,11,13,14]).In this paper, we will investigate the growth of Laplace-Stieltjes transform of infinite order by using a class of functions to reduce the growth of   (, ) or (, ) which is different with the previous form.Thus, we should give the definition of the new function as follows.
Definition 10.If the Laplace-Stieltjes transformation () of infinite order satisfies lim sup where () ∈ F, then  * is called the -order of the Laplace-Stieltjes transform ().
Remark 12.In addition, -order is more precise than order.In fact, for (≥2) being a positive integer, we can find out function () ∈ F and a positive real function () satisfying lim sup For example, letting () = exp  {( log ) 1/ }, () = (log  )  , where  is a finite positive real constant and 0 <  < 1.
The following lemma is very important to study the growth of analytic functions represented by Laplace-Stieltjes transforms convergent on the right half plane, which show the relation between   (, ) and (, ) of such functions.

The Proof of Theorem 4. (⇐):
We consider two steps in this case as follows.
By using the same argument as in the above proof of the theorem, we can prove the necessity of the theorem easily.
Thus, we complete the proof of Theorem 4.

Proofs of Theorems 6 and 7
Similar to the definition of -order of Laplace-Stieltjes transformations in the right half plane, we will give the definition of -order of Laplace-Stieltjes transformations in the level half-strip as follows.
To prove Theorems 6 and 7, we need some lemmas as follows.(55)