On Some Growth Properties of Entire Functions Using Their Maximum Moduli Focusing th Relative Order

Abstract

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

1. Introduction, Definitions, and Notations

Let be an entire function defined in the open complex plane and let be its maximum modulus function. If is nonconstant then is strictly increasing and continuous and its inverse exists and is such that 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.

Definition 1. The order and the lower order of an entire function are defined as

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 then we write and .

Also for and we, respectively, denote and by and .

In this connection we just recall the following definition.

Definition 2 (see [2]). An entire function is said to have index-pair , , if and is not a nonzero finite number, where if and if . Moreover if , then Similarly for , one can easily verify that

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 .

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.

Definition 3 (see [5]). Let and be any two positive integers with . The relative th order of with respect to is defined by If , , and then . If then .

Sánchez Ruiz et al. [6] gave a more natural definition of relative th order of an entire function in light of index-pair which is as follows.

Definition 4. Let and be any two entire functions with index-pairs and , respectively, where 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 .

2. 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 . then

Lemma 2 (see [7]). Let be entire and let be a transcendental entire function of finite lower order. Then, for any ,

Lemma 3 (see [8]). If and are any two entire functions with . then, for any ,

Lemma 4 (see [9]). If and are any two entire functions then for all sufficiently large values of

3. 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 where is a transcendental entire function and are any two positive integers. Then .

Proof. Given that is of finite relative th order with respect to , we have from Definition 4, for a suitable finite number 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 , 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 . Then we have from (22) for all sufficiently large values of that where stands for the constant expression, . Then
Case II. Let us now assume . Then we obtain from (22) for all sufficiently large values of that where stands for a bounded quantity. Then Thus the theorem follows from (24) and (26).

In the forthcoming proofs we will assume the natural number to be , the reasonings being easily adapted for .

Theorem 6. Let , , and be any three transcendental entire functions and let and be two positive integers. If, for any with , , and , it holds that the two limits of some of either(i), ,(ii), , or(iii), exist, then .

Proof. (i) The existence of and implies that given any , 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 is an increasing function of ; it follows from Lemma 2, (27), and (28) that given , for a sequence of values of tending to infinity, the following holds: Hence for all sufficiently large values of . Since is arbitrary and it follows that Under (ii) or (iii) a similar argument applies.

Theorem 7. Let , , and be any three transcendental entire functions and let and be two positive integers. If, for any with , , and , it holds that the two limits of either(i), ,(ii), , or(iii), exist, then .

Proof. (i) Given any , for a sequence of values of tending to infinity, we get that and for all sufficiently large values of that Since is a continuous, increasing, and unbounded function of , we get from above for all sufficiently large values of that Also is an increasing function of ; thus from Lemma 2, (32), and (34) it follows that, given that , for a sequence of values of tending to infinity, Therefore Hence
Since is arbitrary and , , it follows that Under (ii) or (iii) a similar argument may be used.

Theorem 8. Let , , and be any three transcendental entire functions such that where and are any two positive integers. If the limit exists in either(i) or(ii), then

Proof. (i) Since is an increasing function of , it follows from Lemmas 2 and 4, given , 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 Similarly from (41) it follows for all sufficiently large values of that Therefore Hence Also from (45) we obtain for all sufficiently large values of that Then the thesis follows from (43), (46), and (47).
(ii) follows with a similar argument.

Theorem 9. Let , , and be any three transcendental entire functions with . If , , and are any three positive integers with , then under any of the following conditions: (i); (ii); (iii).

Proof. (i) If , since is an increasing function of , it follows from Lemma 1, for all sufficiently large values of , that Therefore Then
(ii) Suppose and .
As is an increasing function of , we get from Lemma 2 that given and any , for all sufficiently large values of , Thus Hence Under (iii) a similar argument to (i) applies.

In the line of Theorem 9 one can easily prove the following result.

Theorem 10. Let , , and be any three transcendental entire functions with . If , and are any three positive integers with , then if any of the following facts happens:(i); (ii).

Theorem 11. Let , , and be any three transcendental entire functions such that . If , , and are any three positive integers with and any of the following two facts happens(i) or(ii), then

Proof . (i) Since the result follows from Theorem 9.
(ii) The proof can be carried out in the line of (i) and Theorem 10.

4. Conclusion

After modifying the notion of relative order of higher dimensions in case of entire functions in [6], where a number of examples of relative order between functions were provided, in this paper we have obtained some growth properties of composite entire functions on the basis of relative th order and relative th lower order. In this process, Theorem 5 and the first part of Theorem 6 and Theorems 7 and 8 can be regarded as extensions of some results of [10].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referee for his/her useful comments.