International Journal of Mathematics and Mathematical Sciences
Volume 2018 (2018), Article ID 7293160, 8 pages
https://doi.org/10.1155/2018/7293160

Research Article

-Growth of an Entire GASHE Function and the Coefficient =

Mohammed Harfaoui, Abdellah Mourassil, and Loubna Lakhmaili

University Hassan II Mohammedia, Laboratory of Mathematics, Criptography and Mechanical F. S.T., BP 146, Mohammedia 20650, Morocco

Correspondence should be addressed to Mohammed Harfaoui; mharfaoui04@yahoo.fr

Received 31 March 2018; Revised 12 September 2018; Accepted 17 September 2018; Published 3 October 2018

Academic Editor: Ram N. Mohapatra

Copyright © 2018 Mohammed Harfaoui 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 extend the work concerning the measures of growth of an entire function solution of the generalized axially symmetric Helmholtz equation ,  , by studying the general measures of growth (-order, lower -order, -type, and lower -type) in terms of coefficients and the ratios of these successive coefficients.

1. Introduction

The partial differential equation

is called generalized axially symmetric Helmholtz equation (GASHE) and the solutions of (1) are called GASHE functions. The GASHE function is regular about the origin and has the following Bessel-Gegenbauer series expansion:

where and , are Bessel functions of first kind, are Gegenbauer polynomials, and .

When series (2) converges absolutely and uniformly on the compact subsets of the whole complex plane, then the GASHE function is said to be entire. For being entire, it is known [1, page 214] that

The concept of order and lower order of an entire function was introduced by R. P. Boas [2] as follows:

and the concept of type and lower type has been introduced to give more precise description of growth of entire functions when they have the same nonzero finite order. An entire function, of order , , is said to be of type and lower type if

where M(r,f) =

Gilbert and Howard [3] have studied the order of an entire GASHE function in terms of the coefficients occurring in the series expansion (2) of . McCoy [4] studied the rapid growth of entire function solution of Helmholtz equation using the concept of index. Kumar [5, 6] extended and improved this result and studied the growth using the concept of index pair. Khan and Ali [7] studied the generalized order and type of entire GASHE function. Kumar and Singh [8] have studied the lower order and lower type of entire GASHE function in terms of the coefficients in its Bessel-Gegenbauer series expansion (2) when the order of is a finite nonzero number. But, for the class of order and , we cannot define a type of . For this reason, numerous attempts have been made to refine the concept of order and type. Therefore, the -order and -type of an entire function have been defined [9, 10]. In this paper, we extend the work of Kumar and Singh [8] to this new classification of entire function.

For and , we define the -order and lower -order as

where and are integers such that where if and if .

The -type and lower -type are defined as

and and for and we use the notations

and

We note that the smallest integer is () since, for example, the order is given by .

To prove that , , or we can prove that for the different values of and . From [9], we define the relation between -order, lower -order, the coefficients of u, and the ratios of these successive coefficients as follows.

Theorem 1. Let be an entire function of -order , and then

where

Theorem 2. Let be an entire function of -order , and then

where

Theorem 3. Let be an entire function of -order and a nondecreasing function of for and then

where

Theorem 4. Let be an entire function of -order and a nondecreasing function of for and then

where

From [10], we define the relation between -type, lower -type, the coefficients of u, and the ratios of these successive coefficients as follows.

Theorem 5. Let be an entire function. The function is of -order and -type if and only if , where if and if , and is defined as

with if and if

Theorem 6. Let be an entire function of -order and lower -type and a nondecreasing function of for and then , where

with if and if

Theorem 7. Let be an entire function of -order and lower -type , and forms a nondecreasing function of for ; then

where

and

with if and if .

2. Auxiliary Results

Let and be two functions defined as

According to [3], we know that if is an entire GASHE function, then and are also entire functions of the complex variable , and

where and . In this section, we shall prove some auxiliary results which will be used in the sequel.

Lemma 8. Let and be entire functions of particular form defined above. Then the -orders and the -types of and , respectively, are identical.

Proof. Let be an entire function, and then, according to Theorem 1, the -order of is given as

and the -type is defined in view of Theorem 5 as

In the consequence of [3], we have

Here we consider the case when .
We have

and then and .
This implies that we will necessarily have to define . And we have

Hence, for the function we have

and

Since and have the same -order it follows that

Now we will prove that and have the same -type for .

In the same way we prove that

Now, for the case , we have

The same is true for

since and have identical -order and -type.

Lemma 9. For an entire GASHE function of -order , lower -order , -type , and lower -type . If and are entire functions as defined above, then

Proof. Using (24), we get

From the above relation we obtain

and since it proves (37) and (38).
Denoting by the common value of -order of , , and , we have from (24)

This proves (39) and (40).

Before we start the next section, let us define , , and .

It is known, according to [3], that if is a nondecreasing function of then and also is a nondecreasing function of .

3. Main Results

Theorem 10. Let be an entire GASHE function of -order and -type . If is a nondecreasing function of for , then

Proof. For an entire function and according to Theorem 1, we have

We know that if is a nondecreasing function of for , and then also and .
Applying (46) to , we get

Similarly, applying (46) to , we prove

Then result (44) is found from the two relations and above and relation (37).
Let be the common -order of and .
The -type of is defined according to Theorem 5 as

and we can easily prove that

and

Equation (45) now follows in view of (37) and (39).
Hence the proof is completed.

Theorem 11. Let be an entire GASHE function of -order , and is a nondecreasing function of for . Then

Proof. For an entire function , according to Theorem 2,

provided is a nondecreasing function of for .
Applying this equation on we get

Since as then

By the same way, we prove

Relation (52) now follows on using (37). Hence the proof is completed.

Theorem 12. Let be an entire GASHE function of -order , lower -order , and lower -type and let be a nondecreasing function of for . Then

Proof. For an entire function , and according to Theorem 3,

We know that if form a nondecreasing function of for , then, also, and .
Applying (59) to , we get

Similarly, applying (59) to , we prove that

Then result (57) is now followed by (38) and above by the two relations for and .
If is an entire function of -order and lower -type , and if is a nondecreasing function of for , then, according to Theorem 6, we have

We can easily prove that

and

Equation (58) now follows in view of (37) and (40). Hence the proof is completed.

Theorem 13. Let be an entire GASHE function of lower -order , and let be a nondecreasing function of for . Then

Proof. Let be an entire function and form a nondecreasing function of for , and, according to Theorem 4, we have

Applying this on and , we can easily prove that

and

Thus, we obtain relation (65) by using (38) and the two equalities above.

Theorem 14. Let be an entire GASHE function of -order , , -type , and lower -type . Then

Proof. If is an entire function of -type and lower -type , then in view of Theorem 7 we have

Applying inequality (71) to , we get

and then

and applying (70) to the function we get

and we can easily prove

and, thus,

and thus the proof is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

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