Abstract
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, , and , which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.
1. Introduction
Our main aim of this paper is to investigate some problems about the growth and approximation of entire functions represented by Laplace–Stieltjes transforms which converge on the whole complex plane. Consider Laplace–Stieltjes transformswhere is a bounded variation on any finite interval . For Laplace–Stieltjes transform (1), setwhere the sequence satisfiesifsimilar to the method in [1], and by using the Valiron–Knopp–Bohr formula, then it yields , that is, is an entire function in the whole plane. We denote to be a class of all the functions of form (1) which are analytic in the half plane , and the sequence satisfies (3) and (5), and denote to be the class of all the functions of form (1) which are analytic in the whole plane , and the sequence satisfies (3), (5), and (4). Thus, if and , then .
By Widder [2], if is absolutely continuous, then is the classical Laplace integral form:if is a step-function and a sequence satisfies (3), andthen becomes a Dirichlet series:where 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 (6) or (8) (see [2]).
In 1963, Yu [1] first proved the Valiron–Knopp–Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of Laplace–Stieltjes transform when .
Theorem 1. If the sequence satisfies (3) and (5), then Laplace–Stieltjes transform (1) satisfieswhere is called the abscissa of uniformly convergence of , and
In addition, Yu [1] introduced the concept of the order of to estimate the growth of the maximal molecule and the maximal term and gave the relation between the Borel line and the order of entire functions represented by Laplace–Stieltjes transform converge in the whole complex plane. After his wonderful works, considerable attention has been paid to the growth, and the value distribution of the functions represented by Laplace–Stieltjes transform or Dirichlet series converges in the half plane or the whole complex plane in the field of complex analysis (see [3–24]).
Set
In view of as , in order to estimate the growth of precisely, we usual use the concepts of order and type as follows.
Definition 1. If andthen it is said that is of order in the whole plane, where . Furthermore, if , the type of is defined byLet be of order in the whole plane and , and let be a nonnegative, continuous, and monotonous function, and it has a left-hand derivative and right-hand derivative in every such thatthen is called the proximate order; furthermore, ifthen will be called the proximate type of with respect to proximate order .
Set , and let and be two reciprocally inverse functions. Then, we obtain the following theorem.
Theorem 2. If Laplace–Stieltjes transform and is of proximate order and order , thenwhereFurthermore, if the sequence satisfiesthen
In order to estimate the growth of Laplace–Stieltjes transform of order precisely, we will use some concepts of the -order as follows.
Definition 2. If , is a positive integer, andthen we call is of -order in the whole plane, where for .
Let be of -order in the whole plane and , similar to the proximate type of with respect to proximate order ; we can give the proximate type of with respect to proximate order of -order as follows. Let be a nonnegative, continuous, and monotonous function, and it has a left-hand derivative and right-hand derivative in every such thatthen is called the -proximate order; furthermore, ifthen will be called the proximate type of with respect to -proximate order .
Set , and let and be two reciprocally inverse functions. Then, we obtain the following theorem.
Theorem 3. If Laplace–Stieltjes transform and is of -proximate order and , , thenwhereFurthermore, if the sequence satisfies (18), then
The other purpose of this article is to study the approximation of the entire function represented by Laplace–Stieltjes transform converges in the whole plane. When Laplace–Stieltjes transform (1) satisfies and for , then will be called an exponential polynomial of degree usually denoted by , i.e., . When we choose a suitable function , the function may be reduced to a polynomial in terms of , that is, . We also use to denote the class of all exponential polynomial of degree almost , that is,
For , we denote by the error in approximating the function by exponential polynomials of degree in uniform norm aswhere
Recently, Singhal and Srivastava [15] and Xu and Liu [25] studied the approximation on Laplace–Stieltjes transforms of finite order and obtained the following theorems.
Theorem 4 (see [15]). If Laplace–Stieltjes transform and is of order and of type , then for any real number , we have
Theorem 5 (see [25], Theorem 2.5). If the Laplace–Stieltjes transform and is of the lower order and if , then for any real number , we have
Furthermore, there exists a positive integer such thatforms a nondecreasing function of for , and then we havei.e.,
Corresponding to Theorems 2 and 3, we obtain the following results concerning the approximation.
Theorem 6. If Laplace–Stieltjes transform and is of proximate order and order , then for any real number , we havewhereFurthermore, if the sequence satisfies (18), then
Theorem 7. If Laplace–Stieltjes transform and is of -proximate order and , , then for any real number , we havewhereFurthermore, if the sequence satisfies (18), then
2. Some Lemmas
Lemma 1. For the functions , and , we have
Proof. Since andthen (40) follows. Moreover,Next, we prove the last equality. Since and , thus, it follows as and , and . Since , then it yieldsTherefore, this completes the proof of Lemma 1.
Remark 1. Obviously, the functions and have the same conclusions as and in Lemma 1.
Lemma 2. Let ; then, the functioncan obtain the minimum
Proof. LetThen, it means . In view of the first equality in (40), we deduce , that is,We can see that, as the value of increases to the given value above, the value of changes from a negative value to a positive value. Thus, can obtain minimumTherefore, this completes the proof of Lemma 2.
Lemma 3. Let ; then, the functioncan obtain the maximum
Proof. LetThen, it follows from (40) that . In view of the first equality in (40), we deduce , that is,It is easy to see that obtain the maximumTherefore, this completes the proof of Lemma 3.
Lemma 4. If Laplace–Stieltjes transform , for any and , we havewhere and are constants.
Proof. We will adapt the method as in Yu [1] and Kong and Hong [7]. SetFor , it followsThen, for any and any , it yieldswhich impliesOn the contrary, for any , it follows that there exists a positive integer such that . Thus, it followsSet ; then, for any real number and , it follows . Thus, for any and , it yields andIn view of (5), there exists a constant such that for . So, we can deduceIn view of (5), for the above and sufficiently large , we have ,and is convergent. Therefore, this completes the proof of Lemma 4.
3. Proofs of Theorems 2 and 3
3.1. The Proof of Theorem 2
We firstly prove . Assume that . Fromthen for any positive number and sufficiently large , it yields
Hence, in view of Lemma 1, we can deducethat is,
Since is arbitrary, we conclude from the above inequality thatand as .
Next, we prove . Assume that ; then, for any , there exists a constant such that
Thus, from Lemma 3, Lemma 4, and (70), it yieldswhich implies
Since is arbitrary, let in (72); it follows that and as .
Therefore, (16) is proved.
Furthermore, if the sequence satisfies (18), then for any , there exists a positive integer such that for . Thus, for any positive integer , it follows
Hence,
Since is arbitrary, then in view of (74), we concludethat is, . Thus, in view of (16), it follows .
Therefore, this completes the proof of Theorem 2.
3.2. The Proof of Theorem 3
We firstly prove . Assume that . Fromthen for any positive number and sufficiently large , it yieldswhere . By Lemmas 2 and 4, for sufficiently large , it followswhere . In view of Lemma 2, it is easy to see that can attain the minimum when satisfies the following equation:
By Lemma 1, we have as . Thus, it follows from (79) that
Hence, the function can obtain the minimumas satisfies equation (80). From and Lemma 1, it yields
So, we can deduce from (78), (81), and (82) thatthat is,
Since is arbitrary, we conclude from the above inequality thatand as .
Next, we prove . Assume that ; then, for any , there exists a constant such that
Thus, from Lemma 3, Lemma 4, and (86), it yields
By using the same argument as in Lemma 3, we can conclude that the functioncan attain the maximumas satisfies
Thus, by Lemma 1, we can deduce from (87) and (89) that
Since is arbitrary, let in (89); it follows that and as . Hence, we prove (23).
Furthermore, if the sequence satisfies (18), then we see from the proof of Theorem 2. Thus, it is easy to prove that from (23).
Therefore, this completes the proof of Theorem 3.
4. Proofs of Theorems 6 and 7
4.1. The Proof of Theorem 6
Firstly of all, we prove . Assume that . By using the same argument as in the proof of Theorem 2, for any positive number and sufficiently large , we have (64). Since , then we have for any . Hence, from the definitions of and , it follows
Let
Then, . Sincethen we can deduce
Hence, from the above inequalities, we conclude
Thus, for any , from (92) and (96) and by combining , it yields
Choosing such that for , thus, for , we can deduce from (97) thatthat is,where is a constant.
Thus, it follows from (65) and (99) that
In view of (100) and using the same argument as in the proof of Theorem 2, we conclude from the above inequality thatand as .
Next, we prove . Assume that ; then, for any , there exists a constant such that
And sincethus, it meansfor any . In addition, there exists such that
Thus, for any and , from (105) and (104), it yields
Hence, in view of (102) and (106) and by using the same argument as in the proof of Theorem 2, we concludewhich implies and as . Hence, (34) holds.
Furthermore, if the sequence satisfies (18), from the proof of Theorem 2, it follows , which means .
Therefore, this completes the proof of Theorem 6.
4.2. The Proof of Theorem 7
By using the same argument as in the proof of Theorem 3 and combining (99), (102), and (106), we can prove the conclusions of Theorem 7 easily.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
H. Y. Xu was responsible for conceptualization and writing and preparing the original draft; H. Y. Xu and W. J. Tang contributed to writing the review and editing; and W. J. Tang, H. Y. Xu, and J. Chen were responsible for funding acquisition.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11561033), the Natural Science Foundation of Jiangxi Province in China (nos. 20181BAB201001 and 20151BAB201008), and the Foundation of Education Department of Jiangxi (nos. GJJ190876, GJJ190895, and GJJ191042) of China.