Growth Analysis of Composite Entire Functions Related to Slowly Changing Functions Oriented Relative Order and Relative Type

Abstract

Some results on comparative growth properties of maximum terms and maximum moduli of composite entire functions on the basis of relative -order and relative -type are proved in this paper.

1. Introduction, Definitions, and Notations

We denote by the set of all finite complex numbers. Let be an entire function defined in the open complex plane . The maximum term of on is defined by and the maximum modulus of on is defined as .

Let be a positive continuous function increasing slowly, that is, as for every positive constant . Singh and Barker [1] defined it in the following way.

Definition 1 (see [1]). A positive continuous function is called a slowly changing function if, for   (>0), and uniformly for   (≥1).
If, further, is differentiable, the above condition is equivalent to

Somasundaram and Thamizharasi [2] introduced the notions of -order and -type for entire function where is a positive continuous function increasing slowly, that is, as for every positive constant “”. The more generalized concepts for -order and -type for entire functions are -order and -type. Their definitions are as follows.

Definition 2 (see [2]). The -order and the -lower order of an entire function are defined as where for and .
Using the inequalities   {cf. [3]}, for  , one may verify that

Definition 3 (see [2]). The -type of an entire function is defined as

If an entire function is nonconstant then is strictly increasing and continuous and its inverse exists and is such that .

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

Datta and Maji [6] gave an alternative definition of relative order and relative lower order in terms of maximum term of an entire function with respect to another entire in the following way.

Definition 4 (see [6]). The relative order and relative lower order of an entire function with respect to an entire function are defined as follows:

In the line of Somasundaram and Thamizharasi [2] and Bernal [4], one may define the relative -order of an entire function in the following manner.

Definition 5 (see [7, 8]). The relative -order and relative -lower order of an entire function with respect to another entire function are defined as

Datta et al. [9] also gave an alternative definition of -order and relative -lower order in terms of maximum term of an entire function which are as follows.

Definition 6 (see [9]). The relative -order and the relative -lower order of an entire function with respect to are as follows:

To determine the relative growth of two entire functions having same nonzero finite relative -order with respect to another entire function, one may introduce the concept of the relative -type in the following way.

Definition 7. The relative -type of an entire function with respect to is defined as follows:

Considering , one may easily verify that Definition 7 coincides with the classical Definition 3.

In the paper we study some comparative growth properties of maximum term and maximum modulus of composition of entire functions corresponding to its left or right factors on the basis of relative -order and relative -type. We do not explain the standard definitions and notations in the theory of entire functions as those are available in [10].

2. Lemmas

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

Lemma 8 (see [11]). If and are any two entire functions then, for all sufficiently large values of ,

Lemma 9 (see [12]). Let and be any two entire functions. Then, for every and ,

Lemma 10 (see [12]). Let and be any two entire functions with . Then for all sufficiently large values of ,

Lemma 11 (see [4]). Suppose is an entire function and , . Then, for all sufficiently large ,

Lemma 12 (see [6]). If is entire and , , then, for all sufficiently large ,

Lemma 13 (see [13]). Let and be any two entire functions. Then for any

3. Theorems

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

Theorem 14. Let and be any two entire functions such that is finite and positive. Also let be an entire function with finite nonzero order. Then, for each and for ,

Proof. If then the theorem is trivial. So we take . Now taking in Lemma 9 and in view of Lemma 12 we have for all sufficiently large values of that Since is an increasing function of , it follows from above for all sufficiently large values of that Again we have for a sequence of tending to infinity and for   (>0), So from (20) and (21) we get for a sequence of tending to infinity that Let Then from (22) we obtain for a sequence of tending to infinity where ,  , , and are finite.
Since , therefore where we choose   (>0) such that This proves the theorem.

Theorem 15. Let and be any two entire functions with is finite. Also let be an entire function with finite nonzero order.Then for each and for , where.

We omit the proof of Theorem 15 as it follows from Theorem 14 and the following inequality in place of (21) for a sequence of values of tending to infinity.

In the line of Theorems 14 and 15, the following two theorems can be proved by using Lemmas 8 and 11 and hence their proofs are omitted.

Theorem 16. Let and be any two entire functions such that is finite and positive. Also let be an entire function with finite nonzero order.Then, for each and for ,

Theorem 17. Let and be any two entire functions with is finite. Also let be an entire function with finite nonzero order. Then, for each and for , where.

Remark 18. In Theorems 14 and 16, if we take the condition “” instead of “ is finite and positive” the theorems remain true with “limit” in place of “limit inferior”.

Remark 19. In Theorems 15 and 17 if we consider the condition “” instead of “” the theorems remain true with “limit” in place of “limit inferior”.

Theorem 20. Let , and be three entire functions such that , , and . Then

Proof. In view of Lemmas 10 and 12 we have Since is an increasing function of , it follows from above for all sufficiently large values of that Now from (34) it follows for a sequence of values of tending to infinity that Now we get for all sufficiently large values of that Hence from (35) and (36) it follows for all sufficiently large values of that Since   (>0) is arbitrary, it follows from (37) that This proves the theorem.

In the line of Theorem 20, the following theorem can be proved.

Theorem 21. Let ,  , and be any three entire functions with , , and . Then

The proof is omitted.

Theorem 22. Let ,  , and be any three entire functions such that , , and . Then

Theorem 23. Let ,  , and be any three entire functions with , , and . Then

We omit the proofs of Theorems 22 and 23 because those can be carried out in the line of Theorems 20 and 21, respectively, and with the help of Lemmas 8 and 11.

Theorem 24. Let ,  , and be any three entire functions such that and . Then

Proof. Let us suppose that the conclusion of the theorem does not hold. Then we can find 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 Thus from (43) and (44) we have for a sequence of values of tending to infinity that This is a contradiction.
This proves the theorem.

Remark 25. Theorem 24 is also valid with “limit superior” instead of “limit” if is replaced by and the other conditions remain the same.

In the line of Theorem 24 the following theorem can also be proved.

Theorem 26. Let , and be any three entire functions with and . Then Further, if instead of then

Corollary 27. Under the assumptions of Theorem 24 or Remark 25 and Theorem 26,

Proof. By Theorem 24 or Remark 25 we obtain, for all sufficiently large values of and for , from which the first part of the corollary follows.
Similarly from Theorem 26, the second part of the corollary is established.

Theorem 28. Let , and be any three entire functions such that ,  ,  , and . Then, for any ,

Proof. From the definition of relative -type and and in view of Lemmas 12 and 13 we obtain for all sufficiently large values of that Also we obtain for a sequence of values of tending to infinity that Now from (51) and (56) it follows for a sequence of values of tending to infinity that In view of the condition , we get from (57) that As   (>0) is arbitrary, it follows from above that Again from (53) and (55), we get for a sequence of values of tending to infinity that Since , we obtain from (60) that As   (>0) is arbitrary, it follows from above that Thus the theorem follows from (59) and (62).

In the line of Theorem 28, we may state the following theorem without its proof.

Theorem 29. Let ,  , and be any three entire functions with ,  ,  , and . Then for any

Theorem 30. Let ,  , and be any three entire functions such that ,  ,  , and . Then

Theorem 31. Let ,  , and be any three entire functions with ,  ,  , and . Then

The proofs of Theorems 30 and 31 are omitted because those can be carried out in the line of Theorems 28 and 29, respectively.

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.