Journal of Function Spaces
Volume 2019 (2019), Article ID 9191346, 12 pages
https://doi.org/10.1155/2019/9191346

Research Article

The Growth on the Maximum Modulus of Double Dirichlet Series

Yong-Qin Cui,¹ Hong-Yan Xu,¹ and Na Li²

¹Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China
²Department of Basic Course, Nanchang Health School, Nanchang, Jiangxi 330006, China

Correspondence should be addressed to Hong-Yan Xu; xhyhhh@126.com

Received 24 October 2018; Accepted 20 December 2018; Published 10 January 2019

Academic Editor: Mitsuru Sugimoto

Copyright © 2019 Yong-Qin Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is to investigate the growth of several entire functions represented by double Dirichlet series of finite logarithmic order, -order. Besides, we also study some properties on the maximum modulus of double Dirichlet series and its partial derivative. Our results are extension and improvement of previous results given by Huo and Liang.

1. Introduction and Basic Notes

For Dirichlet series

where

( are real variables); are nonzero complex numbers.

It is an interesting topic to study some properties of Dirichlet series in the fields of complex analysis;particularly, considerable attention has been paid to the analytic function and entire funtcions represented by Dirichlet series in the half-plane and whole plane, and a number of interesting and important results can be found in [1]. For example, J. R. Yu, G. Srivastava, P. V. Filevich, Z. S. Gao, D. C. Sun, etc. studied the growth and value distribution of Dirichlet series and random Dirichlet series (see [2–17]); Y. Y. Kong, G. T. Deng investigated the growth of Dirichlet-Hadamard product (see [18, 19]); A. R. Reddy, M. N. Sheremeta, C. F. Yi, and H. Y. Xu studied the approximation of Dirichlet series (see [20–24]), and so on.

In 1962, J. R. Yu [25] had made some pioneering research for the growth of double Dirichlet series as follows:

where , , and

However, there were few results about double Dirichlet series because the research involves complex two-dimensional space. In 2009, J. Liu and Z. S. Gao [26] discussed the problem on -order of entire function represented by the double Dirichlet series in the double horizontal line; recently, G. N. Gao [27] further studied the problem about -order of double Dirichlet series by using the Knopp-Kojima method. In this paper, we further investigate the growth of entire functions represented by double Dirichlet series, such as the logarithmic order, -order, and some properties of the maximum modulus of double Dirichlet series and its partial derivatives.

If double Dirichlet series satisfies

then we call that is analytic in the double whole plane, i.e., the entire Dirichlet series of double whole planes. Let be the set of all entire function represented by double Dirichlet series (3) satisfying (4)-(5). Let

be the maximum modulus of in . From the definition of the maximum modulus, if is nonconstant with respect to , we have

and

To state our results, we can introduce the following definitions.

Definition 1. Suppose ; let be the set of with finite logarithmic order and satisfy the following conditions:

(i)For any fixed value of , there exists such that

(ii)For any fixed value of , there exists such that

where are constants, so there exists such that

Definition 2. Suppose that ; for any small and fixed value , there exists such that

and there exists at least a real value and sufficiently large number such that

then we say that has partial logarithmic order with respect to ; similarly, for any small and fixed value , there exists such that

and there exists at least a real value and sufficiently large number such that

then we say that has partial logarithmic order with respect to .

Remark 3. If has partial logarithmic order with respect to , that is,

and if has partial logarithmic order with respect to , that is,

Definition 4. Let , , and satisfy the following conditions:

(i);(ii) has partial logarithmic order with respect to and partial logarithmic order with respect to ;(iii)for any small number , there exists such that

then we say that has finite logarithmic order .

For the logarithmic order of double Dirichlet series , we have the following.

Theorem 5. Let be of finite logarithmic order , ; then

To estimate the growth of more precisely, we will give the logarithmic type of as follows.

Definition 6. Suppose ; let be the set of given by double Dirichlet series of finite logarithmic order has finite logarithmic type and satisfy the following conditions:

(i)For any fixed value of , there exists such that

(ii)For any fixed value of , there exists such that

where , are constants, so there exists such that

Definition 7. Suppose that is of finite logarithmic , and for any small and fixed value , there exists such that

and there exist at least a real value and sufficiently large number such that

then we say that has partial logarithmic type with respect to ; similarly, for any small and fixed value , there exists such that

and there exists at least a real value and sufficiently large number such that

then we say that have partial logarithmic type with respect to .

Remark 8. If is of finite logarithmic order having partial logarithmic type with respect to , that is,

and if has partial logarithmic order with respect to , that is,

Definition 9. Let be of finite logarithmic order , , and satisfy the following conditions:

(i);(ii) has partial logarithmic type with respect to and partial logarithmic type with respect to ;(iii)for any small number , there exists such that

then we say that has finite logarithmic type .

For the logarithmic type of double Dirichlet series , we have the following.

Theorem 10. Let be of finite logarithmic order , , and of finite logarithmic type ; then

In this paper, we also deal with the growth of double Dirichlet series of finite order and infinite order by using a class of functions to reduce which is better than the previous form. So, we firstly give the definition of -order of double Dirichlet series as follows, which is an extension of [6, 9].

Let be the class of all functions satisfying the following conditions:

(i) is defined on and is positive, strictly increasing, and differentiable and tends to as ;(ii) as for , and , , where and .

Similar to the above definitions, we give the -order of double Dirichlet series as follows.

Definition 11. Suppose and ; let be the set of with finite -order, and satisfies the following conditions:

(i)For any fixed value of , there exists such that

(ii)For any fixed value of , there exists such that

where are constants, so there exists such that

where is the inverse function of , especially as .

Definition 12. Suppose that ; for any small and fixed value , there exists such that

and there exist at least a real value and sufficiently large number such that

then we say that has partial -order with respect to ; similarly, for any small and fixed value , there exists such that

and there exist at least a real value and sufficiently large number such that

then we say that has partial -order with respect to .

Remark 13. If has partial -order with respect to , that is,

and if has partial -order with respect to , that is,

Definition 14. Let , , and satisfy the following conditions:

(i);(ii) has partial -order with respect to and partial -order with respect to ;(iii)for any small number , there exists such that

then we say that has finite -order .

For -order of double Dirichlet series, we have the following.

Theorem 15. Let and be of finite -order , ; then

In fact, when , then -order is called finite order. Thus, we obtain the following conclusion.

Corollary 16. Let be of finite order , ; then

And when and , then -order is called the -order. So, we have the following.

Corollary 17. Let be of finite -order , ; then

Similarly, we give the -type of double Dirichlet series as follows.

Definition 18. Suppose and ; let be the set of given by double Dirichlet series of finite logarithmic order having finite -order, and satisfies the following conditions:

(i)For any fixed value of , there exists such that

(ii)For any fixed value of , there exists such that

where are constants, so there exists such that

Definition 19. Suppose that ; for any small and fixed value , there exists such that

and there exist at least a real value and sufficiently large number such that

then we say that has partial -type with respect to ; similarly, for any small and fixed value , there exists such that

and there exist at least a real value and sufficiently large number such that

then we say that has partial -type with respect to .

Remark 20. If has partial -type with respect to , that is,

and if has partial -order with respect to , that is,

Definition 21. Let be of finite -order , , and satisfy the following conditions:

(i);(ii) has partial -type with respect to and partial -type with respect to ;(iii)for any small number , there exists such that

then we say that has finite -type .

For -type of double Dirichlet series , we have the following.

Theorem 22. Let and be of finite -order , , and of finite -type ; then

Particularly, we can get the following corollaries.

Corollary 23. Let and be of finite order and of finite type ); then

Corollary 24. Let and be of finite -order , , and of finite -type as ; then

Remark 25. In 2010 and 2015, Liang and Gao [28–30] further investigated the growth and convergence of -multiple Dirichlet series. We only listed some Liang’s definitions as ; Liang defined the order of multiple Dirichlet series by

and if , then the type of multiple Dirichlet series

Obviously, our definitions about the order and type of double Dirichlet series are more general than Liang’s.

The other purpose of this paper is to investigate some relation between the partial derivatives and growth of . In order to state our results, we first give some notations as follows. Let

and

for .

Theorem 26. Let be of finite logarithmic order , ; then

and

Theorem 27. Let be of finite logarithmic order , ; then

Theorem 28. Let be of finite -order , ; then

and

Theorem 29. Let be of finite -order , ; then

For finite order and finite -order, some corollaries are obtained below.

Corollary 30. Let be of finite order , ; then

and

Corollary 31. Let be of finite order , ; then

Corollary 32. Let be of finite -order , , ; then

and

Corollary 33. Let be of finite -order , ,; then

2. Proofs of Theorems 5–22

2.1. The Proof of Theorem 5

From (18), let ; it follows

When , let ; then

Next, we continue to prove that .

Case 1. Suppose that ; then there exist two constants such that and . From (18), there exist two sequences , such that

that is,

Hence, when is fixed at and , we have

Since , we can obtain a contradiction with the assumption of Theorem 5 from the above inequality.

Case 2. Suppose that ; then there exist two constants such that and . From (75), there exists such that for

for , that is,

Then it follows

Thus, we can choose such that

then for , we have

If , we can choose such that and

that is,

as .
Since is arbitrary and , from (83) and (85), we can get a contradiction with the assumptions of being of finite logarithmic order .
Therefore, this completes the proof of Theroem 5 from Cases 1 and 2.

2.2. The Proof of Theorem 10

The proof of Theorem 10 is similar to the argument as in Theorem 5; in order to facilitate the readers, we still give the proof of Theorem 10 as follows.

From (29), take ; then

Let ; then

Thus we have

Now, we are going to prove that .

Case 1. Suppose that ; then there exist two constants satisfying and . Thus, from (88), there exist two sequences such that

i.e.,

then when is fixed at and , it yields

Since , we can get a contradiction with the assumptions of being of finite logarithmic type .

Case 2. Suppose that ; then there exist two constants satisfying and . From (88), there exists such that

that is,

From (93), there exists such that

We choose such that

Then, for , it follows

If , we can choose such that for

that is,

Since is arbitrary and , in view of (96) and (98), it yields a contradiction with the assumption of being of finite logarithmic type .

Therefore, we completes the proof of Theorem 10 from Cases 1 and 2.

2.3. Proofs of Theorems 15 and 22

Similar to the same argument as in the proofs of Theorems 5 and 10, we can complete the proofs of Theorems 15 and 22 easily.

3. Proofs of Theorems 26 and 27

3.1. The proof of Theorem 26

For a fixed value of , let

it can easily be shown that is monotonic increasing for Set such that and ; then we have

it follows

Similar, there exists such that

From (101) and (102), we have

Since is of finite logarithmic order , , it follows from (103) and (104) that

and

On the other hand, for any fixed value of , is an increasing convex function of ; then

Hence, from (107) and the assumptions of Theorem 26, we can easily get

Similarly,

Therefore, the conclusion of Theorem 26 follows from (105), (106), (108), and (109).

Thus, this completes the proof of Theorem 26.

3.2. The Proof of Theorem 27

Since is of finite logarithmic order , , from Theorem 26, we have that for any , and , there exists a real number such that

and for any , there exists a real number such that

Thus, there exists a real number such that

for . Then it follows

Here we prove that . If , there exists two constants such that and

for . Thus, from (114), we obtain that, for any , there exists a real number such that

Hence, since , from Theorem 26, we can easily obtain a contradiction with the assumption that is of finite partial logarithmic order with respect to .

Thus, ; this completes the proof of Theorem 27.

4. Proofs of Theorems 28 and 29

From the definition of , it follows that

for any finite constant . Moreover, we have the following.

Lemma 34 (see [6, 9, 24]). Let and be the nonconstant function satisfying as and

if satisfies Then we have

Remark 35. Under the assumptions of Lemma 34, for , we have

as .

4.1. The Proof of Theorem 28

By using the same argument as in the proof of Theorem 26, and combining with Lemma 34, we can easily prove Theorem 28.

4.2. The Proof of Theorem 29

Since is of finite -order , , from Theorem 28, we have that for any , and , there exists a real number such that

and for any , there exists a real number such that

Since is strictly increasing and is the inverse function of , then is also strictly increasing. Thus, there exists a real number such that

and

for . Thus, we have

that is,

for . By Lemma 34, it follows

Here is to prove that . If , there exist two constants such that and

for . Thus, from (127), we obtain that, for any , there exists a real number such that

Hence, since , from Theorem 28, we can easily obtain a contradiction with the assumption that is of finite partial -order with respect to .

Thus, ; this completes the proof of Theorem 29.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest in the manuscript.

Authors’ Contributions

Hong-Yan Xu contributed to conceptualization; Yong-Qin Cui and Hong-Yan Xu contributed to writing-original draft preparation; Hong-Yan Xu contributed to writing-review and editing; Yong-Qin Cui, Hong-Yan Xu, and Na Li contributed to funding acquisition. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), and the Foundation of Education Department of Jiangxi (GJJ170759, GJJ170788) of China.

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