Research Article | Open Access
-Growth of an Entire GASHE Function and the Coefficient =
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.
The partial differential equationis 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 .
The concept of order and lower order of an entire function was introduced by R. P. Boas  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 ifwhere M(r,f) =
Gilbert and Howard  have studied the order of an entire GASHE function in terms of the coefficients occurring in the series expansion (2) of . McCoy  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  studied the generalized order and type of entire GASHE function. Kumar and Singh  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  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 notationsand
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 , 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 thenwhere
Theorem 2. Let be an entire function of -order , and thenwhere
Theorem 3. Let be an entire function of -order and a nondecreasing function of for and thenwhere
Theorem 4. Let be an entire function of -order and a nondecreasing function of for and thenwhere
From , 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 aswith if and if
Theorem 6. Let be an entire function of -order and lower -type and a nondecreasing function of for and then , wherewith if and if
Theorem 7. Let be an entire function of -order and lower -type , and forms a nondecreasing function of for ; thenwhereandwith if and if .
2. Auxiliary Results
Let and be two functions defined as
According to , we know that if is an entire GASHE function, then and are also entire functions of the complex variable , andwhere 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 asand the -type is defined in view of Theorem 5 asIn the consequence of , we haveHere 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 thatNow, for the case , we have The same is true forsince 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
Before we start the next section, let us define , , and .
It is known, according to , 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 haveWe 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 asand we can easily prove thatandEquation (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 proveRelation (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