On Some Growth Properties of Entire Functions Using Their Maximum Moduli Focusing (p, q)th Relative Order

We discuss some growth rates of composite entire functions on the basis of the definition of relative (p, q)th order (relative (p, q)th lower order) with respect to another entire function which improve some earlier results of Roy (2010) where p and q are any two positive integers.


Introduction, Definitions, and Notations
We use the standard notations and definitions in the theory of entire functions which are available in [1]. In the sequel we use the following notation: The following definitions are well known. log [2] ( ) log , Juneja et al. [2] defined the ( , )th order and ( , )th lower order of an entire function , respectively, as follows: where , are any two positive integers with ≥ . If = and = 1 then we write ( , 1) = [ ] and Also for = 2 and = 1 we, respectively, denote (2, 1) and (2, 1) by and .

2
The Scientific World Journal In this connection we just recall the following definition.
Definition 2 (see [2]). An entire function is said to have index-pair ( , ), An entire function for which ( , )th order and ( , )th lower order are the same is said to be of regular ( , )-growth. Functions which are not of regular ( , )-growth are said to be of irregular ( , )-growth.
Bernal [3] introduced the definition of relative order of with respect to , denoted by ( ) as follows: The definition coincides with the classical one [4] if = exp.
Similarly one can define the relative lower order of with respect to denoted by ( ) as follows: In the case of relative order, it therefore seems reasonable to define suitably the relative ( , )th order of entire functions. Lahiri and Banerjee [5] also introduced such definition in the following manner.
Sánchez Ruiz et al. [6] gave a more natural definition of relative ( , )th order of an entire function in light of indexpair which is as follows. and , , and are all positive integers such that ≥ and ≥ . Then the relative ( , )th order of with respect to is defined as Similarly one can define the relative ( , )th lower order of an entire function with respect to another entire function denoted by ( , ) ( ) where and are any two positive integers in the following way: In fact Definition 4 improves Definition 3 ignoring the restriction ≥ .
In this paper we wish to prove some results related to the growth rates of entire functions on the basis of relative ( , )th order and relative ( , )th lower order with respect to another entire function extending some earlier results for any two positive integers and .

Lemmas
In this section we present some lemmas which will be needed in the sequel.
Lemma 1 (see [7]). If and are any two entire functions with (0) = 0. then The Scientific World Journal 3 Lemma 2 (see [7]). Let be entire and let be a transcendental entire function of finite lower order. Then, for any > 0, Lemma 3 (see [8]). If and are any two entire functions with (0) = 0. then, for any 0 < < 1, Lemma 4 (see [9]). If and are any two entire functions then for all sufficiently large values of ⩾ 0

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

Theorem 5.
Let be an entire function and let be any polynomial such that ∘ has got finite relative ( , )th order with respect to ℎ where ℎ is a transcendental entire function and , are any two positive integers. Then Proof. Given that ∘ is of finite relative ( , )th order with respect to ℎ, we have from Definition 4, for a suitable finite number > 0 and for all sufficiently large values of , that Now let be the order of the polynomial so that Then by Cauchy's inequality we get from (18) that Now given 0 < < 1, in view of Lemma 3 and from (17) it follows for all sufficiently large values of that We rewrite the above to the equivalent for all sufficiently large values of that Therefore from (21) we get for all sufficiently large values of that Case I. Assume = 1. Then we have from (22) for all sufficiently large values of that where (1) stands for the constant expression, log(( | |) −1 2 ). Then Case II. Let us now assume > 1. Then we obtain from (22) for all sufficiently large values of that where (1) stands for a bounded quantity. Then i.e., Thus the theorem follows from (24) and (26).
In the forthcoming proofs we will assume the natural number to be > 1, the reasonings being easily adapted for = 1.

Theorem 6. Let , , and ℎ be any three transcendental entire functions and let and be two positive integers. If, for any
, with 0 < < 1, > 0, and ( + 1) > 1, it holds that the two limits , ∈ R + of some of either  The Scientific World Journal Proof. (i) The existence of and implies that given any > 0, for sufficiently large values of , Since ( ) is a continuous, increasing, and unbounded function of , we get from above for all sufficiently large values of that Also −1 ℎ ( ) is an increasing function of ; it follows from Lemma 2, (27), and (28) that given > 0, for a sequence of values of tending to infinity, the following holds: i.e., Hence lim sup for all sufficiently large values of . Since > 0 is arbitrary and ( + 1) > 1 it follows that Under (ii) or (iii) a similar argument applies.
Theorem 7. Let , , and ℎ be any three transcendental entire functions and let and be two positive integers. If, for any , with > 1, 0 < < 1, and > 1, it holds that the two limits , ∈ R + of either Proof. (i) Given any > 0, for a sequence of values of tending to infinity, we get that and for all sufficiently large values of that i.e., Since ( ) is a continuous, increasing, and unbounded function of , we get from above for all sufficiently large values of that Also −1 ℎ ( ) is an increasing function of ; thus from Lemma 2, (32), and (34) it follows that, given that > 0, for a sequence of values of tending to infinity, The Scientific World Journal 5 Therefore Hence lim sup Since > 0 is arbitrary and > 1, > 1, it follows that Under (ii) or (iii) a similar argument may be used.
Proof. (i) Since −1 ℎ ( ) is an increasing function of , it follows from Lemmas 2 and 4, given > 0, for all sufficiently large values of , that respectively. Therefore from (40) we get for all sufficiently large values of that From here it follows that lim sup  (45) i.e.,