Generalized Relative Type and Generalized Weak Type of Entire Functions

Abstract

We study some relative growth properties of entire functions with respect to another entire function on the basis of generalized relative type and generalized relative weak type.

1. Introduction

A single valued function of one complex variable which is analytic in the finite complex plane is called an integral (entire) function. For example , , and are examples of entire functions. In the value distribution theory one studies how an entire function assumes some values and the influence of assuming certain values in some specific manner on a function. In Rolf Nevanlinna initiated the value distribution theory of entire functions. This value distribution theory is a prominent branch of complex analysis and is the prime concern of the paper. Perhaps the Fundamental Theorem of Classical Algebra which states that “if is a polynomial of degree with real or complex coefficients, then the equation has at least one root” is the most well known value distribution theorem.

The value distribution theory deals with various aspects of the behavior of entire functions one of which is the study of comparative growth properties. For any entire function , the maximum modulus of is the function defined as

Similarly function is defined for another entire function . The ratio as evaluates the growth of with respect to in terms of their maximum moduli.

However, the order of an entire function which is generally used in computational purpose is defined in terms of the growth of with respect to function as

Bernal [1, 2] introduced the relative order between two entire functions to avoid comparing growth just with , extending the notion of relative order as cit.op. Lahiri and Banerjee [3] introduced the definition of generalized relative order. In the case of generalized relative order, it therefore seems reasonable to define suitably the generalized relative type (generalized relative weak type) of an entire function with respect to another entire function in order to compare the relative growth of two entire functions having the same nonzero finite generalized relative order (generalized relative lower order) with respect to another entire function. In this connection Datta et al. [4] introduced the definition of generalized relative type (generalized relative weak type) of an entire function with respect to another entire function.

For entire functions, the notions of the growth indicators such as order and type (weak type) are classical in complex analysis and, during the past decades, several researchers have already been exploring them in the area of comparative growth properties of composite entire functions in different directions using the classical growth indicators. But, at that time, the concepts of relative order (generalized relative orders), relative type (generalized relative type), and relative weak type (generalized relative weak type) of entire functions and their technical advantages of not comparing with the growth of are not at all known to the researchers of this area. Therefore the studies of the growth of composite entire functions in the light of their relative order (generalized relative orders), relative type (generalized relative type), and relative weak type (generalized relative weak type) are the main concern of this paper. In fact some light has already been thrown on such type of works by Datta et al. in [4–8]. Actually, in this paper, we study some relative growth properties of entire functions with respect to another entire function on the basis of generalized relative type and generalized relative weak type.

2. Notation and Preliminary Remarks

Our notations are standard within the theory of Nevanlinna’s value distribution of entire functions and therefore we do not explain those in detail as available in [9]. In the sequel the following two notations are used:

Taking this into account, Juneja et al. [10] defined the th order and th lower order of an entire function , respectively, as follows:where , are any two positive integers with .

These definitions extended the definitions of order and lower order of an entire function which are classical in complex analysis for integers and since these correspond to the particular case and . Further, for and , the above definitions reduce to generalized order [11] (resp., generalized lower order ).

In this connection let us recall that if , then the following properties hold: Similarly, for , one can easily verify that

Recalling that for any pair of integer numbers , the Kronecker function is defined by for and for , the aforementioned properties provide the following definition.

Definition 1 (see [10]). An entire function is said to have index-pair if . Otherwise, is said to have index-pair , , if and

Definition 2 (see [10]). An entire function is said to have lower index-pair if . Otherwise, is said to have lower index-pair , , if and

Remark 3. An entire function of index-pair is said to be of regular -growth if its th order coincides with its th lower order, otherwise is said to be of irregular -growth.

To compare the growth of entire functions having the same th order, Juneja et al. [12] also introduced the concepts of th type and th lower type in the following manner.

Definition 4 (see [12]). The th type and the th lower type of entire function having finite positive th order    are defined as where , are any two positive integers, if , and for

Remark 5. For and , the above definitions reduce to generalized type and generalized lower type of an entire function Moreover, when and , then and are correspondingly denoted as and which are, respectively, known as type and lower type of entire

Now we introduce the following definitions in order to determine the relative growth of two entire functions having the same nonzero finite th lower order in the following manner.

Definition 6. The th weak type and the growth indicator of an entire function having finite positive th lower order    are defined by where , are any two positive integers, if , and for

Remark 7. If we consider and in the above definitions, then the growth indicators and are correspondingly denoted as and . Further, for and , the above definition reduces to the classical definition as established by Datta and Jha [13]. Also and stand for and

For any two entire functions and , Bernal [1, 2] initiated the definition of relative order of with respect to , indicated by , as follows:which keeps away from comparing growth just with to find out order of entire functions as we see earlier and of course this definition corresponds with the classical one [14] for

Remark 8. In line with the above definition, one may define the relative lower order of with respect to , denoted by , as

Extending this notion, Lahiri and Banerjee [3] gave a more generalized concept of relative order in the following way.

Definition 9 (see [3]). If is a positive integer, then the th generalized relative order of with respect to , denoted by , is defined by Clearly and .

Remark 10. Likewise one can define the generalized relative lower order of with respect to denoted by as

Moreover to compare the relative growth of two entire functions having the same nonzero finite generalized relative order with respect to another entire function, Datta et al. [4] introduced the definition of generalized relative type and generalized relative lower type of an entire function with respect to another entire function, which are as follows.

Definition 11 (see [4]). Assume that and are entire functions with The generalized relative type and generalized relative lower type of with respect to are defined as

For , Definition 11 reduces to classical definition as established by Roy [15].

Further, to determine the relative growth of two entire functions having the same nonzero finite generalized relative lower order with respect to another entire function, Juneja et al. [10] introduced the concepts of generalized relative weak type and growth indicator of an entire function with respect to another entire function in the following manner.

Definition 12 (see [4]). The generalized relative weak type and the growth indicator of an entire function with respect to another entire function having finite positive generalized relative lower order are defined as

Remark 13. For , Definition 12 reduces to the classical definition as established by Datta and Biswas [8].

3. Lemmas

In this section we present two lemmas which will be needed in the sequel.

Lemma 14 (see [16]). Let be an entire function with and let be entire of regular -growth, where and are positive integers such that . Then

Lemma 15 (see [16]). Let be an entire function with growth and let be entire with , where and are positive integers such that Then

4. Main Results

In this section we present the main results of the paper.

Theorem 16. Let be any entire function with and let be any entire function with index-pairs , where and are positive integers such that Also let be of regular -growth. Then

Proof. Fix . From the definitions of and we have for all sufficiently large values of thatand also for a sequence of values of tending to infinity we get thatSimilarly, from the definitions of and , it follows for all sufficiently large values of thatand for a sequence of values of tending to infinity we obtain thatNow, from (20) and in view of (22), we get for a sequence of values of tending to infinity thatAs is arbitrary, in view of Lemma 14, it follows thatAnalogously from (19) and in view of (25) it follows for a sequence of values of tending to infinity thatSince is arbitrary, we get from the above and Lemma 14 thatAgain in view of (23) we have from (18), for all sufficiently large values of , that Since is arbitrary, we obtain in view of Lemma 14 thatAgain, from (19) and in view of (22), we get for all sufficiently large values of thatAs is arbitrary, it follows from the above and Lemma 14 thatAlso, in view of (24), we get from (18) that, for a sequence of tending to infinity,Since is arbitrary, we get from Lemma 14 and the above thatSimilarly, from (21) and in view of (23), it follows for a sequence of values of tending to infinity thatAs is arbitrary, we obtain from Lemma 14 and the above thatThus the theorem follows from (27), (29), (31), (33), (35), and (37).

In view of Theorem 16, one can easily verify the following corollaries.

Corollary 17. Let be an entire function such that and let be an entire function of regular -growth, where and are all positive integers such that Then

Corollary 18. Let be an entire function with and let be entire function of perfectly regular -growth, where and are positive integers with Then In addition, if , then

Corollary 19. Let be an entire function with Then, for any entire function ,(i) when ,(ii) when ,(iii) when ,(iv) when ,where is any positive integer with and is of regular -growth.

Corollary 20. Let be an entire function with regular -growth where are positive integers with Then, for any entire function with , one has(i) when ,(ii) when ,(iii) when ,(iv) when .

In line with Theorem 16 and with the help of Lemma 15, one can prove the following theorem, and therefore its proof is omitted.

Theorem 21. Let be any entire function with and let be any entire function with index-pairs , where and are positive integers such that Then

In view of Theorem 21, one can easily derive the following corollaries.

Corollary 22. Let be an entire function with and and let be an entire function with index-pairs , where and are positive integers such that . Then

Corollary 23. Let be an entire function with and let be an entire function with . ThenIn addition, if , then

Corollary 24. Let be an entire function with index-pairs , where and are positive integers such that . Then, for any entire function with , one has(i) when ,(ii) when ,(iii) when ,(iv) when .

Corollary 25. Let be an entire function with Then for any entire function with index-pairs , where and are positive integers such that , one has(i) when ,(ii) when ,(iii) when ,(iv) when .

Similarly, in line with Theorems 16 and 21 and with the help of Lemmas 14 and 15, one may easily prove the following two theorems, and therefore their proofs are omitted.

Theorem 26. Let be any entire function with and let be any entire function with index-pairs , where and are positive integers such that Also let be of regular -growth. Then

In view of Theorem 26, the following corollaries may also be obtained.

Corollary 27. Let be an entire function such that and let be an entire function of regular -growth, where and are positive integers such that Then

Corollary 28. Let be an entire function with and let be an entire function of regular -growth, where and are positive integers such that with Then In addition, if , then

Corollary 29. Let be an entire function with and let be an entire function of regular -growth, where and are positive integers such that Then(i) when ,(ii) when ,(iii) when ,(iv) when .

Corollary 30. Let be an entire function with regular -growth, where and are all positive integers such that Then, for any entire function with , one has(i) when ,(ii) when ,(iii) when ,(iv) when .

Theorem 31. Let be any entire function with and let be any entire function with index-pairs , where and are positive integers such that Then

From Theorem 31 the following corollaries are immediate.

Corollary 32. Let be an entire function with and and let be an entire function with index-pairs , where and are positive integers such that and Then

Corollary 33. Let be an entire function with regular growth and let be an entire function with , where and are positive integers such that ThenIn addition, if , then

Corollary 34. Let be an entire function with index-pairs , where and are positive integers such that and Then, for any entire function with , one has(i) when ,(ii) when ,(iii) when ,(iv) when .

Corollary 35. Let be an entire function with Then, for any entire function with index-pairs , where and are positive integers such that , one has(i) when ,(ii) when ,(iii) when ,(iv) when .

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.