Growth Rates of Meromorphic Functions Focusing Relative Order

Abstract

A detailed study concerning some growth rates of composite entire and meromorphic functions on the basis of their relative orders (relative lower orders) with respect to entire functions has been made in this paper.

1. Introduction, Definitions and Notations

Let be a meromorphic and let be an entire function defined in the open complex plane and . 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 and meromorphic functions those are available in [1, 2].

To start our paper we just recall the following definitions.

Definition 1. The order (the lower order ) of an entire function is defined as If is a meromorphic function, one can easily verify that where is the Nevanlinna's characteristic function of (cf. [1]).

Bernal [3, 4] introduced the definition of relative order of an entire function with respect to another entire function denoted by as follows:

The definition coincides with the classical one [5] 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 [6] introduced the definition of relative order of a meromorphic function with respect to an entire function in the following way.

Definition 2 (see [6]). Let be any meromorphic function and let 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. [6]) that if , then Definition 2 coincides with the classical definition of the order of a meromorphic function .

In this paper we wish to prove some results related to the growth rates of composite entire and meromorphic functions on the basis of relative order (relative lower order) of meromorphic functions with respect to an entire function.

2. Lemmas

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

Lemma 1 (see [7]). Let be meromorphic and let be entire and suppose that . Then for a sequence of values of tending to infinity,

Lemma 2 (see [8]). Let be meromorphic and let be entire such that and . Then for a sequence of values of tending to infinity, where .

Lemma 3 (see [9]). Let be a meromorphic function and let be an entire function such that and . Then for a sequence of values of tending to infinity,

Lemma 4 (see [9]). Let be a meromorphic function of finite order and let be an entire function such that . Then for a sequence of values of tending to infinity,

3. Theorems

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

Theorem 1. Let be a meromorphic function and let , be any two entire functions such that for any satisfying , , , and . Then

Proof. From we have for all sufficiently large values of that and from we obtain for all sufficiently large values of that Also is an increasing function of it follows from (13) and (14) and Lemma 1 for a sequence of values of tending to infinity that that is, that is, that is, that is, that is, Since is arbitrary and , it follows from above that This proves the theorem.

In the line of Theorem 1 and with the help of Lemma 2 one may state the following theorem without its proof.

Theorem 2. Let be a meromorphic function and let , be any two entire functions such that and . Further suppose that for any , , satisfying , , and . Then

Theorem 3. Let be a meromorphic function and let , be any two entire functions such that for any , satisfying , , and . Then

Proof. From we have for all sufficiently large values of that and from we obtain for all sufficiently large values of that As is an increasing function of , it follows from (26) and (27) and Lemma 1 for a sequence of values of tending to infinity that
Since is arbitrary and , , the theorem follows from above.

Theorem 4. Let be a meromorphic function and let , be any two entire functions such that and . Further suppose that for any satisfying , , , and . Then

We omit the proof of Theorem 4 as it can be carried out in the line of Theorem 3 and with the help of Lemma 2.

Theorem 5. Let be a meromorphic function and let , be any two entire functions such that and . Then

Proof. Suppose .
As is an increasing function of , we get from Lemma 1 for a sequence of values of tending to infinity that Thus the theorem follows.

In the line of Theorem 5, one can easily prove the following theorem.

Theorem 6. Let be a meromorphic function and let , be any two entire functions such that and . Then

Theorem 7. Let be a meromorphic function and let , be any two entire functions such that and . Then

Proof. In view of Theorem 5, we obtain that Thus the theorem follows.

Theorem 8. Let be a meromorphic function and let , be any two entire functions such that and . Then

The proof of Theorem 8 is omitted as it can be carried out in the line of Theorem 7 and in view of Theorem 6.

Theorem 9. Let be a meromorphic function and let be an entire functions such that . Also let be an entire function with nonzero order. Then for every positive constant and every real number ,

Proof. If is such that , then the theorem is trivial. So we suppose that .
Since is an increasing function of , we get from Lemma 1 for a sequence of values of tending to infinity that where we choose .
Again from the definition of , it follows for all sufficiently large values of that Now from (38) and (39), it follows for a sequence of values of tending to infinity that Since as , the theorem follows from above.

In the line of Theorem 9 and with the help of Lemma 2, one may state the following theorem without its proof.

Theorem 10. Let be a meromorphic function with nonzero finite lower order and let be an entire function with nonzero finite order. Also let be an entire function such that and . Then for every positive constant and every real number ,

Theorem 11. Let be a meromorphic function and let be an entire function with nonzero order. Also let be an entire function such that and . Then for every positive constant and every real number ,

Theorem 12. Let be a meromorphic function with nonzero finite lower order and let be an entire function with nonzero finite order. Also let be an entire function such that . Then for every positive constant and every real number ,

We omit the proof of Theorems 11 and 12 as those can be carried out in the line of Theorems 9 and 10, respectively.

Theorem 13. Let be a meromorphic function with nonzero finite order and lower order. Also let , be any two entire functions such that . Then for every positive constant and each ,

Proof. If , then the theorem is obvious. We consider .
Since is an increasing function of , it follows from Lemma 3 for a sequence of values of tending to infinity that Again for all sufficiently large values of we get that Hence for a sequence of values of tending to infinity, we obtain from (45) and (46) that So from (47) we obtain that This proves the theorem.

Theorem 14. Let be a meromorphic function with nonzero finite order and lower order. Also let , be any two entire functions such that and . Then for every positive constant and each ,

The proof is omitted as it can be carried out in the line of Theorem 13.

Theorem 15. Let be a meromorphic function with finite order and let be an entire function with nonzero finite lower order. Also let be another entire function such that and . Then for every positive constant and each ,

Theorem 16. Let be a meromorphic function with finite order and let be an entire function with nonzero finite lower order. Also let be another entire function such that . Then for every positive constant and each ,

We omit the proofs of Theorems 15 and 16 as those can be carried out in the line of Theorems 13 and 14, respectively, and with the help of Lemma 4.

Theorem 17. Let be a meromorphic function and let , be any two entire functions such that and . Then for every ,

Proof. If possible, let there exist a constant such that for a sequence of values of tending to infinity, Again from the definition of , it follows for all sufficiently large values of that Now combining (53) and (54), we have for a sequence of values of tending to infinity that which contradicts the condition .
So for all sufficiently large values of we get that from which the theorem follows.

Remark 18. Theorem 16 is also valid with “limit superior” instead of “limit” if is replaced by and the other conditions remain the same.

Corollary 19. Under the assumptions of Theorem 16 and Remark 18, respectively hold.

Proof. By Theorem 16, we obtain for all sufficiently large values of and for that from which the first part of the corollary follows.
Similarly using Remark 18, we obtain the second part of the corollary.

Analogously one may state the following theorem and corollaries without their proofs as those can be carried out in the line of Remark 18, Theorem 16, and Corollary 19, respectively.

Theorem 20. If is a meromorphic function and , are any two entire functions such that and , then for every ,

Corollary 21. Theorem 17 is also valid with “limit” instead of “limit superior” if is replaced by and the other conditions remain the same.

Corollary 22. Under the assumptions of Theorem 17 and Corollary 21, respectively hold.

4. Conclusion

The notion of order and lower order which are the main tools to study the comparative growth properties of entire and meromorphic functions is very much classical in complex analysis. On the basis of the order and lower order of entire or meromorphic functions, several researchers have already explored their works on the area of comparative growth rates of composite entire and meromorphic functions in different directions. In fact, the main aim of this paper is actually to generalize the growth estimates of composite entire and meromorphic functions on the basis of their relative orders and relative lower orders with respect to an entire function. Moreover, the treatment of these notions may also be done by taking the help of slowly changing functions in order to study some different growth properties of composite entire and meromorphic functions.

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for his/her valuable suggestions towards the improvement of the paper.