#### 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 serieswhere ( 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 satisfiesthen 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). Letbe the maximum modulus of in . From the definition of the maximum modulus, if is nonconstant with respect to , we haveand

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 thatwhere are constants, so there exists such that

*Definition 2. *Suppose that ; for any small and fixed value , there exists such thatand there exists at least a real value and sufficiently large number such thatthen we say that has partial logarithmic order with respect to ; similarly, for any small and fixed value , there exists such thatand there exists at least a real value and sufficiently large number such thatthen 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 thatthen 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 thatwhere , 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 thatand there exist at least a real value and sufficiently large number such thatthen we say that has partial logarithmic type with respect to ; similarly, for any small and fixed value , there exists such thatand there exists at least a real value and sufficiently large number such thatthen 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 thatthen 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 thatwhere are constants, so there exists such thatwhere is the inverse function of , especially as .

*Definition 12. *Suppose that ; for any small and fixed value , there exists such thatand there exist at least a real value and sufficiently large number such thatthen we say that has partial -order with respect to ; similarly, for any small and fixed value , there exists such thatand there exist at least a real value and sufficiently large number such thatthen 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 thatwhere are constants, so there exists such that

*Definition 19. *Suppose that ; for any small and fixed value , there exists such thatand there exist at least a real value and sufficiently large number such thatthen we say that has partial -type with respect to ; similarly, for any small and fixed value , there exists such thatand there exist at least a real value and sufficiently large number such thatthen 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 byand 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. Letandfor .

Theorem 26. *Let be of finite logarithmic order , ; thenand*

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

Theorem 28. *Let be of finite -order , ; thenand*

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 , ; thenand*

Corollary 31. *Let be of finite order , ; then*

Corollary 32. *Let be of finite -order , , ; thenand*

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 followsWhen , 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 thatthat is,Hence, when is fixed at and , we haveSince , 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 forfor , that is,Then it followsThus, we can choose such thatthen for , we haveIf , we can choose such that andthat 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 ; thenLet ; then Thus we have