Generalized Relative Type and Generalized Weak Type of Entire Functions

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.


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 exp , sin , and cos 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 1926 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 ( ) = 0 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 exp function as Bernal [1,2] introduced the relative order between two entire functions to avoid comparing growth just with exp , 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 2 Journal of Complex Analysis 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 exp 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][5][6][7][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.

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 = 2 and = 1 since these correspond to the particular case (2, 1) = and (2, 1) = . Further, for = and = 1, the above definitions reduce to generalized order [ ] [11] (resp., generalized lower order [ ] ).
In this connection let us recall that if 0 < ( , ) < ∞, then the following properties hold:   Recalling that for any pair of integer numbers , the Kronecker function is defined by , = 1 for = and , = 0 for ̸ = , the aforementioned properties provide the following definition.
Definition 1 (see [10]). An entire function is said to have index-pair Definition 2 (see [10]). An entire function is said to have 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.
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.

Remark 7.
If we consider = and = 1 in the above definitions, then the growth indicators ( , 1) and ( , 1) are correspondingly denoted as [ ] and [ ] . Further, for = 2 and = 1, the above definition reduces to the classical definition as established by Datta and Jha [13]. Also and stand for [2] and [2] .
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 exp to find out order of entire functions as we see earlier and of course this definition corresponds with the classical one [14] for = exp .

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.
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.
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.

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

Main Results
In this section we present the main results of the paper.
and also for a sequence of values of tending to infinity we get that Similarly, from the definitions of ( , ) and ( , ), it follows for all sufficiently large values of that Journal of Complex Analysis 5 and for a sequence of values of tending to infinity we obtain that Now, from (20) and in view of (22), we get for a sequence of values of tending to infinity that .
Analogously from (19) and in view of (25) it follows for a sequence of values of tending to infinity that . .
Again in view of (23) we have from (18), for all sufficiently large values of , that .
Again, from (19) and in view of (22), we get for all sufficiently large values of that .
Also, in view of (24), we get from (18) that, for a sequence of tending to infinity, .
Similarly, from (21) and in view of (23), it follows for a sequence of values of tending to infinity that .
In view of Theorem 21, one can easily derive the following corollaries. .
In view of Theorem 26, the following corollaries may also be obtained.

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