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Journal of Complex Analysis
Volume 2013 (2013), Article ID 472170, 6 pages
Measures of Growth of Entire Solutions of Generalized Axially Symmetric Helmholtz Equation
Research and Post Graduate Studies, Department of Mathematics, M.M.H.College, Model Town, Ghaziabad 201001, India
Received 18 August 2012; Accepted 27 October 2012
Academic Editor: Vladislav Kravchenko
Copyright © 2013 Devendra Kumar and Rajbir Singh. 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.
For an entire solution of the generalized axially symmetric Helmholtz equation , measures of growth such as lower order and lower type are obtained in terms of the Bessel-Gegenbauer coefficients. Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients.
The solutions of the partial differential equation are called the generalized axially symmetric Helmholtz equation functions (GASHEs). The GASHE function , regular about the origin, has the following Bessel-Gegenbauer series expansion: where , are Bessel functions of first kind, and are Gegenbauer polynomials. A GASHE function is said to be entire if the series (2) converges absolutely and uniformly on the compact subsets of the whole complex plane. For being entire, it is known [1, page 214] that:
Now we define
Following the usual definitions of order and type of an entire function of a complex variable , the order and type of are defined as
Gilbert and Howard  have studied the order of an entire GASHE function in terms of the coefficients ’s occurring in the series expansion (2) (see also [1, Theorem 4.5.9]). It has been noticed that the coefficients characterizations for lower order and lower type of have not been studied so far. In this paper, we have made an attempt to bridge this gap. McCoy  studied the order and type of an entire function solutions of certain elliptic partial differential equation in terms of series expansion coefficients and approximation errors. Recently, Kumar [4, 5] obtained some bounds on growth parameters of entire function solutions of Helmholtz equation in in terms of Chebyshev polynomial approximation errors in sup-norm. In the present paper, we have considered the different partial differential equation from those of McCoy  and Kumar [4, 5] and obtained the growth parameters such as lower order and lower type of entire GASHE function in terms of the coefficients in its Bessel-Gegenbauer series expansion (2). Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients. Our approach and method are different from all these of the above authors.
2. Auxiliary Results
In this section, we shall prove some preliminary results which will be used in the sequel.
We prove the following lemma.
Lemma 1. If is an entire GASHE function, then for all , , and for all ,
Proof. First we prove right hand inequality. Using the relations
in (2), we get
Now to prove left hand inequality, we use the orthogonality property of Gegenbauer polynomials [1, page 173] and the uniform convergence of the series (2) as
From the series expansion of , we get and for , where denotes the integral part of , we have
From (7), (9), and (11), for , we now get
Since as , we can choose constants and such that for . Thus, for , (12) gives that
Hence the proof of Lemma 1 is complete.
We now define
Lemma 2. If is an entire GASHE function, then and are also entire functions of the complex variable . Further where and .
Lemma 3. Let and be entire functions of particular form defined by (14). Then orders and types of and , respectively, are identical.
Proof. It is well known [6, pages 9–11] that if is an entire function, then the order and type are given as
Hence for the function , we have
Similarly, for , we have
It follows that . Since and have the same order, using (18), we can easily see that . Hence the proof is complete.
In analogy with the definitions of order and type, we define lower order and lower type as
Theorem 4. Let be an entire GASHE function of order , lower order , type , and lower type . If and are entire functions as defined above, then
Proof. Using (15) we get
In view of [6, page 13] for an entire function of finite order we have,
Now from above relation (26), we obtain
Since , it proves (22) and (24).
Denoting by the common value of order of , , and , we have from (15),
Hence by Lemma 3 we obtain (23). Similarly, we can prove (25).
Lemma 5. If forms a nondecreasing function of , then and also form a nondecreasing function of , where .
Proof. We have
By logarithmic differentiation, we get
Let for any .
Hence is a decreasing function and subsequently for . Hence is nondecreasing if is nondecreasing. Similarly we can prove the result for .
3. Main Results
Now we prove the following theorem.
Theorem 6. Let be an entire GASHE function of order , type , and lower type . Then
Further, if forms a nondecreasing function of for all , then
Proof. If is an entire function of order , type , and lower type , then we have [7, Theorem 1]
Applying right hand inequality to , we get
To prove left hand inequality in (34), we consider the entire function . Then
Thus the proof of (34) is complete. To prove (35), consider an entire function of order and type . If forms a nondecreasing function of for , then we know (, [9, Theorem 2]) that
Further, we have [8, Theorem 3]
Now let us suppose that forms a nondecreasing function of for . From Lemma 5, also forms a nondecreasing function of for . Using (40) to , we get
Now using (41) for , we get
Since , , thus, we get
Hence the proof is complete.
Theorem 7. Let be an entire GASHE function of order , , lower order , and lower type . If forms a nondecreasing function of for , then
Proof. For entire function , if forms a nondecreasing function of for , then we have (, [10, Theorem 2])
Let forms a nondecreasing function for . Applying Lemma 5 and (48) to , we get
Similarly, using Lemma 5 and (48) for entire function , we have
The result (46) is now followed by (24) and above two relations for and .
If is an entire function of order , lower type , and forms a nondecreasing function of for , then by a result of Shah , we have Equation (47) now follows in view of (22) and (25). Hence the proof is complete.
Theorem 8. Let be an entire GASHE function of lower order , and let forms a nondecreasing function of for . Then
Proof. For an entire function , from [12, Corollary, page 312], we get
provided forms a nondecreasing function of for . Applying this condition on , we can easily show, as in Theorem 6, that
Applying (53) to also, we have
The relation (48) now follows on using (24). Hence the proof is complete.
The authors are very much thankful to the reviewers for giving fruitful comments to improve the paper. This work is supported by University Grants Commission, New Delhi, India, under the Grant no. F. 11-27-2004 (SA-I).
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