Research Article | Open Access
Growth and Approximation of Laplace-Stieltjes Transform with -Proximate Order Converges on the Whole Plane
One purpose of this paper is to study the growth of entire functions defined by Laplace-Stieltjes transform converges on the whole complex plane, by introducing the concept of -proximate order, and one equivalence theorem of the -proximate order of Laplace-Stieltjes transforms is obtained. Besides, the second purpose of this paper is to investigate the approximation of entire functions defined by Laplace-Stieltjes transforms with -proximate order, and some results about the -proximate order, the error, and the coefficients of Laplace-Stieltjes transforms are obtained, which are generalization and improvement of the previous theorems given by Luo and Kong, Singhal, and Srivastava.
The main purpose of this paper is to investigate the growth and approximation of entire functions represented by Laplace-Stieltjes transforms which convergent on the whole complex plane. For Laplace-Stieltjes transforms,where is a bounded variation on any finite interval and σ and t are real variables. In fact, if is absolutely continuous, then becomes the classical Laplace integral form:
If is a step function, we can choose a sequence satisfyingand letThus, becomes a Dirichlet series:where are real variables and are nonzero complex numbers; if is an increasing continuous function which is not absolutely continuous, then integral (1) defines a class of functions which cannot be expressed either in form (2) or (6).
As we know, the Laplace-Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure; however, it is often defined for functions with values in a Banach space. It can be used in many fields of mathematics, such as functional analysis, and certain areas of theoretical and applied probability.
In 1963, Yu  first proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of the LaplaceStieltjes transform.
Moreover, Yu  introduced the concepts of the order of and estimated the growth of the maximal molecule , the maximal term , the Borel line, and the order of entire functions represented by LaplaceStieltjes transform convergent in the whole complex plane. After his works, considerable attention has been paid to the growth and the value distribution of the functions represented by LaplaceStieltjes transform convergent in the half plane or whole complex plane in the field of complex analysis (see [2–16]).
By using the same argument as in , we can get the similar result about the abscissa of uniformly convergent of easily. Ifby (3), (6), and Theorem 1, we have ; thus, it can be said that is an entire function in the whole plane.
Since and trend to as , in order to estimate the growth of precisely, the concepts of order and type is introduced as follows.
Definition 1. If the LaplaceStieltjes transform (11) satisfies (the sequence satisfy (3), (9), and (20)) andwe call is of order ρ in the whole plane, where . Furthermore, if , the type of is defined by
Remark 1. The concept of type is used to compare the growth of two LaplaceStieltjes transforms of the same nonzero finite order. For example, let , by simple computation, and we have , but and . Thus, we can see that the growth of is faster than as .
Remark 2. Valiron  used intermediate comparison functions called the proximate order to refine these growth scales, which make it unnecessary to consider functions of minimal or maximal type.
Remark 3. However, if and , we cannot estimate the growth of such functions precisely by using the concept of type.
For , Kong  studied the growth of such functions by using the concepts of the Sun’s type function , where , is the decreasing function , , and . In fact, one can also estimate the growth of LaplaceStieltjes transforms by using the concepts of p-order were given by Sato . Ifwhere , and , then is said to be of index p if and . If , we also compare their growth by using the concepts of p-type. However, there are two shortcomings in the concepts of -order and -type as follows. S1: one can hardly compare the growth of such functions of zero order by using the idea of -order. S2: the growth of such functions of and is incomparable by using the idea of -order.For the first shortcoming, there were some concepts such as logarithmic order, h-order, and generalize order (see [21–24]). However, to cover these shortcomings simultaneously, the concepts of -order and -type were good ideas and were used to estimate the growth of a class of entire functions represented by LaplaceStieltjes transforms more precisely, which were given by Juneja, Kapor, and Bajpai [25–27].
Definition 2. If the LaplaceStieltjes transform (11) satisfies (the sequence satisfy (3), (9), and (20)) andthen we say that is of -order in the whole plane, where are two integers such that and , and , , and , .
Remark 4. Throughout this paper, the conditions and are always supposed to hold.
Remark 5. It can be seen that if and if . Furthermore, if , then for , for , and for .
Definition 3. It is said that the LaplaceStieltjes transform is of index-pair , , and if , and is not a finite nonzero number, where if and if . If is of index-pair , then is called its -order.
The concept of -type is introduced below, which is used to compare the growth of such functions of the same -order.
Definition 4. The LaplaceStieltjes transform (11) of -order is said to be of the -type if
Remark 6. From the above definitions, it can be seen easily that the concepts of -order and -type can cover the above shortcomings (S1) and (S2) in some sense.
However, these concepts are inadequate for comparing the growth of LaplaceStieltjes transforms of the same -orders but of infinite -types. Inspired by this question, one purpose of this paper is to improve the above inadequacy and investigate the growth of LaplaceStieltjes transforms.
The structure of this paper is listed as following. Section 2 is to investigate the growth of LaplaceStieltjes transforms with the -proximate order, and an equivalence theorem on the -proximate order is obtained among the -order , -type , , and . Section 3 is to deal with a study of the approximation on LaplaceStieltjes transforms of the -proximate order and gives a relation theorem about the error , -order , -type , and -proximate order. The topic of growth and approximation of LaplaceStieltjes transforms of -proximate order, it seems that this topic has never been treated before. Our results in this paper are some improvements of the previous theorems given by Kong, Luo, and Sun.
2. The -Proximate Order of LaplaceStieltjes Transforms
Next, we introduce the definition of -proximate order of LaplaceStieltjes transforms.
Definition 5. A positive function defined on , , is a proximate order of LaplaceStieltjes transforms (11) with index-pair if satisfies the following conditions:(i) as , , where if and if .(ii) as , where .(iii)ifthen is said to be the -proximate order of the LaplaceStieltjes transform if .
Letwhere if and zero otherwise, by a simple computation, we can get that is a monotone increase function of σ for . Thus, we can define a real function of such thatHere, one main theorem of this paper is as follows.
To prove this theorem, we require the following lemmas.
Proof. Here, two cases will be considered as follows.
Case 1. If , then and , . Thus, it follows that as and , and . Since , it yields
Case 2. If . From Definition 4, we concludeThus, for any given and , from (28) it followsthat is,Hence,For , it is obviously. For , by a similar argument as in (28), we can deducethat is,Hence,Thus, this completes the proof of Lemma 1 from Cases 1 and 2.
Lemma 2 (see , Lemma 2.1). If the L-S transform , for sufficiently larger and , we havewhere C is a constant.
2.1. The Proof of Theorem 2
Here, we first prove the necessity of Theorem 2.
We will divide into two steps to prove the necessity.
Step one. We first prove that
Since for any given and , it follows
Let , then
It thus follows from Lemma 2 that
Now, we will consider three cases as follows.
Case 3. Suppose that and , then . From (39), it followsTake σ such thatthat is,In view of (40) and (43), it yieldsthat is,Since , thus , that is, . From (40), we have and as . Hence, from (43), and combining with ε is arbitrarily, we can deduce
Case 4. Suppose that and , then . From (39), it followsTake σ such thatFrom (46), it follows . Thus, we can conclude from (45) thatthat is,From (46), it yields and as . Hence, from (48), and combining with ε is arbitrarily, we conclude
Case 5. Suppose that , then . Take σ such thatthat is,Substituting (50) into (38), it yieldsThus, it followsthat is,In view of (54), we can deduceSince as and , by Lemma 1, let and in (56), and it yieldsTherefore, we can conclude from (44), (46), and (56) thatwhere H is stated as in Theorem 2.
Step Two. Next, we will prove thatcannot hold. From (58), let to be a real constant such that andThus, for any small ε such that and such that , it followsHere, three cases will be considered as follows.
Case 6. Suppose that and , then and , and (60) becomesThus, it followsthat is,Take σ such thatthat is,Then, in view of Lemma 2 and (63), we can deduceThus, from Lemma 2 and (66) and since , it meansa contradiction.
Case 8. Suppose that , then and . Then, it follows from (60) thatTake σ such thatthat is,Then, from Lemma 1 and (73), it yieldsThus, from Lemma 2 and (76) and since and , we concludea contradiction.
Therefore, the necessity of Theorem 2 is proved from Steps one and two.
In fact, by using the same argument as in the processing of the necessity of Theorem 2, we can easily prove the sufficiency of Theorem 2. Hence, this completes the proof of Theorem 2.
3. The Approximation of LaplaceStieltjes Transforms
We denote to be a class of all the functions of form (9) which are analytic in the half plane and the sequence satisfy (3) and (6), and denote to be the class of all the functions of form (9) which are analytic in the half plane and the sequence satisfy (3), (6), and (11). Thus, if and , then . If the LaplaceStieltjes transform (11), for , and , then will be called an exponential polynomial of degree k usually denoted by , , . When we choose a suitable function , the function may be reduced to a polynomial in terms of , that is, .
For , we denote by the error in approximating the function by exponential polynomials of degree n in uniform norm aswhere
In 2017, Singhal and Srivastava  studied the approximation of LaplaceStieltjes transforms of finite order and obtained the following theorem.
Theorem 3 (see ). If the LaplaceStieltjes transform and is of order and of type T, then for any real number , we have
Recently, the second author investigated the approximation of LaplaceStieltjes transforms of infinite order by using the concepts of X-order and the type function and obtained some theorems as follows.
Theorem 4 (see ). If the LaplaceStieltjes transform , then for any fixed real number , we havewhere .
Theorem 5 (see ). If the LaplaceStieltjes transform , then for any fixed real number , we have
Remark 8. The definitions of X-order and the type function can be found in [30, 31].
The other purpose of this paper is to investigate the approximation of LaplaceStieltjes transforms with proximate order , the result concerning the error , and the proximate order of such function is obtained. The main theorem is listed below.
Theorem 6. If the LaplaceStieltjes transform and is of (p, q) proximate order , then for any real number , we havewhere , ϱ, and H are stated as in Theorem 2.
Proof. Suppose thatthen for any given and , we havewhere . Since , thus for any constant , we have . For , it follows from the definitions of and thatSetThen, we have Sincewhere β is an any real number, it followsWhen , we have . It yields thatThus, from the definition of and