Growth Analysis of Composite Entire and Meromorphic Functions in the Light of Their Relative Orders

Abstract

We study some comparative growth properties of composite entire and meromorphic functions on the basis of their relative orders (relative lower orders).

1. Introduction

Let be meromorphic and be an entire function defined in the open complex plane . The maximum modulus function corresponding to entire is defined as . For meromorphic , cannot be defined as is not analytic. In this situation one may define another function known as Nevanlinna’s characteristic function of , playing the same role as maximum modulus function in the following manner: where the function known as counting function of -points (distinct -points) of meromorphic is defined as Moreover, we denote by the number of -points (distinct -points) of in and an -point is a pole of . In many occasions and are denoted by and , respectively.

The function alternatively denoted by known as the proximity function of is defined as follows: Also we may denote by .

When is an entire function, the Nevanlinna’s characteristic function of is defined as

Further, if is a nonconstant entire function then and are both strictly increasing and continuous functions of . Also their inverses and exist, respectively, and are such that and .

In this connection we just recall the following definition which is relevant.

Definition 1 (see [1]). A nonconstant entire function is said to have the Property (A) if, for any and for all sufficiently large , holds. For examples of functions with or without the Property (A), one may see [1].

However, for any two entire functions and , the ratio as is called the growth of with respect to in terms of their maximum moduli. Similarly, when and are both meromorphic functions, the ratio as is called the growth of with respect to in terms of their Nevanlinna’s characteristic functions. The notion of the growth indicators such as order and lower order of entire or meromorphic functions which are generally used in computational purpose is defined in terms of their growth with respect to the exponential function as the following.

Definition 2. The order (the lower order ) of an entire function is defined as where for and . Further, if is a meromorphic function one can easily verify that

Bernal [1, 2] introduced the definition of relative order of an entire function with respect to another entire function , denoted by to avoid comparing growth just with as follows:

The definition coincides with the classical one [3] if .

Similarly, one can define the relative lower order of an entire function with respect to another entire function denoted by as follows:

Extending this notion, Lahiri and Banerjee [4] introduced the definition of relative order of a meromorphic function with respect to an entire function in the following way.

Definition 3 (see [4]). Let be any meromorphic function and be any entire function. The relative order of with respect to is defined as Likewise, one can define the relative lower order of a meromorphic function with respect to an entire function denoted by as follows:
It is known (cf. [4]) that if then Definition 3 coincides with the classical definition of the order of a meromorphic function .

For entire and meromorphic functions, the notions of their growth indicators such as order and lower order are classical in complex analysis and during the past decades, several researchers have already been exploring their studies in the area of comparative growth properties of composite entire and meromorphic functions in different directions using the classical growth indicators. But at that time, the concepts of relative orders and relative lower orders of entire and meromorphic functions and their technical advantages of not comparing with the growths of are not at all known to the researchers of this area. Therefore the studies of the growths of composite entire and meromorphic functions in the light of their relative orders and relative lower orders are the prime concern of this paper. In fact some light has already been thrown on such type of works by Datta et al. [5]. We do not explain the standard definitions and notations of the theory of entire and meromorphic functions as those are available in [6, 7].

2. Lemmas

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

Lemma 1 (see [8]). Let be meromorphic and be entire; then, for all sufficiently large values of ,

Lemma 2 (see [9]). Let be meromorphic and be entire and suppose that . Then, for a sequence of values of tending to infinity,

Lemma 3 (see [10]). Let be meromorphic and be entire such that and . Then, for a sequence of values of tending to infinity, where .

Lemma 4 (see [11]). Let be an entire function which satisfies the Property (A), , , and . Then

3. Theorems

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

Theorem 1. Let be a meromorphic function and be an entire function with and let be an entire function with finite order. If satisfies the Property (A), then, for every positive constant and each ,

Proof. Let us suppose that and . If , then the theorem is obvious. We consider .
Since is an increasing function of , it follows from Lemma 1, Lemma 4, and the inequality   {cf. [6]} for all sufficiently large values of that Again for all sufficiently large values of we get that Hence for all sufficiently large values of we obtain from (16) and (17) that where we choose .
So from (18) we obtain that This proves the theorem.

Remark 2. In Theorem 1 if we take the condition instead of , the theorem remains true with “limit inferior” in place of “limit.”

In view of Theorem 1 the following theorem can be carried out.

Theorem 3. Let be a meromorphic function and let , be any two entire functions where is of finite order and , . If satisfies the Property (A), then, for every positive constant and each ,

The proof is omitted.

Remark 4. In Theorem 3 if we take the condition instead of , the theorem remains true with “limit” replaced by “limit inferior”.

Theorem 5. Let be a meromorphic function and let , be any two entire functions such that and . Also suppose that satisfies the Property (A). Then, for a sequence of values of tending to infinity,

Proof. Let us consider . Since is an increasing function of , it follows from Lemma 1 that, for a sequence of values of tending to infinity, Now from (17) and (22), it follows for a sequence of values of tending to infinity that As we can choose () in such a way that
Thus from (23) and (24) we obtain that Now from (25), we obtain for a sequence of values of tending to infinity and also for Thus the theorem follows.

In the line of Theorem 5, we may state the following theorem without its proof.

Theorem 6. Let and be any two entire functions with and let be a meromorphic function with finite relative order with respect to . Also suppose that and satisfies the Property (A). Then, for a sequence of values of tending to infinity,

As an application of Theorem 5 and Lemma 2, we may state the following theorem.

Theorem 7. Let be a meromorphic function and let , be any two entire functions such that and . If satisfies the Property (A), then

The proof is omitted.

Similary in view of Theorem 6 and Lemma 3, the following theorem can be carried out.

Theorem 8. Let be a meromorphic function and let , be any two entire functions with , , and . Moreover satisfies the Property (A). Then

The proof is omitted.

Theorem 9. Let be a meromorphic function and let , be any two entire functions with and . Then where .

Proof. Let . As is an increasing function of , it follows from Lemma 2 for a sequence of values of tending to infinity that Again for all sufficiently large values of we get that So combining (31) and (32), we obtain for a sequence of values of tending to infinity that
Since , it follows from (33) that Hence the theorem follows.

Corollary 10. Under the assumptions of Theorem 9,

Proof. In view of Theorem 9, we get for a sequence of values of tending to infinity that from which the corollary follows.

Theorem 11. Let be a meromorphic function and let , be any two entire functions such that (i) , (ii) , and (iii) . Then where .

Proof. From the definition of relative order and relative lower order, we obtain for arbitrary positive and for all sufficiently large values of that Therefore, from (38), it follows for all sufficiently large values of that Thus the theorem follows from (34) and (39).

Similarly, one may state the following theorems and corollary without their proofs as those can be carried out in the line of Theorems 9 and 11 and Corollary 10, respectively.

Theorem 12. Let be a meromorphic function and be an entire function with . Then, for any entire function , where .

Theorem 13. Let be a meromorphic function and let , be any two entire functions such that (i) and (ii) . Then where .

Corollary 14. Under the assumptions of Theorem 12,

Theorem 15. Let be a meromorphic function and let be an entire function with . Then, for any entire function , where and .

Proof. Let . As is an increasing function of , it follows from (31) for a sequence of values of tending to infinity that So for a sequence of values of tending to infinity we get from above that Again we have for all sufficiently large values of that Now combining (45) and (46), we obtain for a sequence of values of tending to infinity that from which the theorem follows.

In view of Theorem 15 the following theorem can be carried out.

Theorem 16. Let be a meromorphic function and let , be any two entire functions with and . Then where and .

The proof is omitted.

Theorem 17. Let be an entire function satisfying the Property (A) and let be a meromorphic function such that . Also let and be any two entire functions with finite nonzero order such that . Then, for every meromorphic function with ,

Proof. Since , we can choose () in such a way that As is an increasing function of , it follows from Lemma 2 for a sequence of values of tending to infinity that Now from the definition of relative order of with respect to we have for arbitrary positive and for all sufficiently large values of that Now for any , we get from (16), (51), (52), and in view of (50) for a sequence of values of tending to infinity that which proves the theorem.

In the line of Theorem 17 the following theorem can be carried out.

Theorem 18. Let be an entire function satisfying the Property (A) and let be a meromorphic function such that . Also let and be any two entire functions with finite nonzero order and also . Then, for every meromorphic function with ,

4. Conclusion

Actually this paper deals with the extension of the works on the growth properties of composite entire and meromorphic functions on the basis of their relative orders and relative lower orders. These theories can also be modified by the treatment of the notions of generalized relative orders (generalized relative lower orders) and th relative orders (th relative lower orders). Moreover, some extensions of the same type may be done in the light of slowly changing functions.

Conflict of Interests

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