#### Abstract

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. Our results exhibit the growth of Laplace-Stieltjes transforms from the point of view of approximation.

#### 1. Introduction and Main Results

In 1946, Widder  considered the convergence of the following form where is a bounded variation on any finite interval , and and are real variables and obtained the following theorem.

Theorem 1 (see (, Theorem 1, Page 36)). If then (1) converges for every for which , and where .
As 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. Moreover, it can be used in many fields of mathematics, such as functional analysis, and certain areas of theoretical and applied probability.
In view of Ref. , can become the classical Laplace integral form when is absolutely continuous. Moreover, if is a step function, choosing a sequence such that then we can conclude from Theorem 1 that becomes a Dirichlet series where is real variables, is nonzero complex numbers. 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. Hence, we can say that Laplace-Stieltjes transform is a general form of Dirichlet series. 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.
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. 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 ). For example, Yu  in 1963 first proved a series of theorems about the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence and uniform convergence of Laplace-Stieltjes transforms, the maximal molecule , the maximal term , the Borel line and the order of entire functions represented by Laplace-Stieltjes transforms convergent in the complex plane. Batty, Sheremeta, Kong, and Sun investigated the growth of analytic functions with kinds of order defined by Laplace-Stieltjes transforms (see ), and Shang, Gao, Zhang, and Xu investigated the value distribution of such functions (see ).
In 2012,Luo and Kong  studied the following form, is differ from (1), of Laplace-Stieltjes transform where is stated as in (1), and satisfies (5) and Set By using the same argument as in , we can get the similar result about the abscissa of uniformly convergent of easily. If by (5), (9)-(12), and Ref., one can get that , , is entire in the whole plane.
Set

Definition 2 (see ). If Laplace-Stieltjes transform (8) satisfies (the sequence satisfy (5) and (9)-(12)), we define the order and the lower order of by respectively, where .

Remark 3. If , and , we say that is an entire function of zero order, finite order, and infinite order in the whole plane, respectively.

Definition 4. If Laplace-Stieltjes transform (8) satisfies (the sequence satisfy (5) and (9)-(12)) and is of order , then we define which is called the type of Laplace-Stieltjes transform .
In 2012 and 2014, Luo and Kong [29, 30] studied the growth of Laplace-Stieltjes transform of finite order and obtained the following theorem.

Theorem 5 (see [29, 30]). If Laplace-Stieltjes transform (8) satisfies (the sequence satisfy (5) and (9)-(12)), and is of order and of type , then In order to state our main results of this paper, we also introduce some definitions and notations below. We denote by the set of all the functions of the form (8) which are analytic in the half plane and the sequence satisfy (5), (9), and (10) and by the set of all the functions of the form (8) which are analytic in the half plane and the sequence satisfy (5) and (9)-(12). Obviously, if and , then . If (8) satisfies for , and , then we say that is an exponential polynomial of degree usually denoted by , , . If we choose a suitable function , the function may be reduced to a polynomial in terms of , that is, . We denote to be the class of all exponential polynomial of degree almost , that is, For , we use to denote the error in approximating the function by exponential polynomials of degree in uniform norm as where Around 2017, Singhal and Srivastava [31, 32] studied the approximation of entire functions represented by Laplace-Stieltjes transforms (8) of finite order and obtained the following result.

Theorem 6 (see ). If Laplace-Stieltjes transform and is of order and of type , then for any real number , we have In the same year, the authors  further the approximation on the entire function represented by Laplace-Stieltjes transforms with irregular growth and obtained.

Theorem 7 (see (, Theorem 6)). If the Laplace-Stieltjes transform and is of the lower order , if , then for any real number , we have Furthermore, there exists a positive integer such that forms a nondecreasing function of for , and then we have As far as we know, there are few papers focusing on the approximation of Laplace-Stieltjes transform of infinite order. Inspired by this issue, our main purpose of this paper is to deal with the approximation of Laplace-Stieltjes transforms of infinite order with the help of the type function given by Sun. In 1986, Sun  studied the existence of type function of the complex function of infinite order and established a new type function which is more precise than Xiong’s.

Theorem 8 (see ). If is a continuous function in and then we say that is the type function of , if there exist two continuous and differential functions and satisfying monotonous, decreasing and trend to 0, monotonous, increasing(ii)(iii)For sufficient large , (iv)where and mean that and a sequence such that .

Remark 9. If and is of infinite order , then in view of Theorem 7, there exists a type function such that where and .

Theorem 10. Let be of infinite order, and the sequence satisfies (5), (9), (12) and and then for any real number , we have where , , and are stated as in Theorem 10.

Remark 11. We can easily get (10) from (26), thus when the sequence satisfies (5), (9), (12), and (26). That is to say, our condition in our theorem is better than the previous results.

Theorem 12. Under the assumptions of Theorem 10, then where Let , and then it follows for any positive integer and any real number . Hence, we get the following corollary.

Corollary 13. Under the assumptions of Theorem 10, then

Theorem 14. Under the assumptions of Theorem 10, then we have

#### 2. The Proof of Theorem 6

To prove Theorem 10, we require the following lemma.

Lemma 15 (see (, Theorem 6b)). If and are continuous and is of bounded variation in , and if then

Lemma 16. If Laplace-stieltjes transform is of infinite order, and the sequence satisfies (5), (9), (12), and (26), then where , , and are stated as in Theorem 7.

Proof. The idea of the proof of this lemma come from Ref. . Next, we will show the completely details.
Set From (9), there exists satisfying and then it follows , as . Thus, for and by Theorem 1 and Lemma 15, we deduce that is, Hence, for any and any , it follows that is, Therefore, we can conclude from (39) that On the other hand, assume that then for any fixed and sufficiently large , it follows For any positive real number , in view of (5), there exists a positive integer such that ; thus, it yields where . Similar to the argument as in (38), it follows Set , where , and then from (44), we have where Thus, for any real number , in view of (26), there exists a positive integer such that Hence, we can conclude from (45) and (47) that Set , where is an integral function. Then, it follows as . So, from (48), we can deduce Hence, from (39), (49), and by Theorem 8, it becomes where is a finite constant. Since is arbitrary and , then we conclude Therefore, this completes the proof of Lemma 2 from (40) and (51).
Proof of Theorem 10. In view of and then it is obvious that the conclusion of Theorem 10 holds as . Next, we will prove that the conclusion of Theorem 10 holds for .
If , then for any fixed real number , there exists a positive integer such that Let and be the inverse function of , and then we know in view of Theorem 8 that is an increasing function for . Thus, from the above inequality, we can deduce that is, For sufficiently large , set and then it follows If , it yields from (55) and (57) that If , that is, , it yields from (55) and (57) that Hence, we can conclude from (58) and (59) that For any , it follows and thus for any , we can deduce by combining the above inequalities that On the other hand, there exists such that Hence, from (60)–(63) and for any , it follows that is,

Since is arbitrary and by Lemma 16, we get

We therefore completes the Proof of Theorem 6.

#### 3. Proofs of Theorems 7 and 8

##### 3.1. The Proof of Theorem 7

Here, we will adopt the reduction to absurdity to prove Theorem 7. Suppose that

If , set . From (65), for any small and any positive integer , we have

If and , we have . Moreover, for and , it follows

Similar to (44), we have

Thus, for any , we can conclude from (39), (67), and (68) that

From (9), there exists such that for . Then, for , it follows from (69) that that is, where is a constant. Hence, we conclude from (66) and (71) that

On the other hand, in view of and , there exists a subsequence such that

We choose a sequence such that and then it follows from (72) and (74) that

Hence,

In view of , then it follows . Thus, by combining (74) and (76), we deduce which is a contradiction with (73).

If , we choose , and by using the same argument as above, we also get a contradiction.

Therefore, this completes the Proof of Theorem 7.

##### 3.2. The Proof of Theorem 8

From Lemma 15, it is easy to get the following lemma.

Lemma 17. If Laplace-Stieltjes transform is of infinite order, and the sequence satisfied (5), (9), (12), and (26), then Proof of Theorem 8. If , then it follows By combining Corollary 13, it yields

which is a contradiction with the properties of . Hence, . Thus, for any fixed and sufficiently large , we have where and are two reciprocally inverse functions; that is, For any fixed sufficiently large , take and then it yields If , from (81) and (83), it follows If , that is, , thus in view of (81) and (83), it yields Hence, from (62), (63), (84), and (85), we deduce Let and by Lemma 17, and it leads to Suppose that Set , and then for any positive integer and sufficiently large , from (71), we have where is a constant. Since , then there exists a subsequence such that We choose a sequence such that Thus, it follows from (89) and (91) that That is, We therefore can conclude from (91) that