Journal of Mathematics

Volume 2014 (2014), Article ID 908798, 18 pages

http://dx.doi.org/10.1155/2014/908798

Research Article

On Iterated Entire Functions with ()th Order

Sanjib Kumar Datta,¹ Tanmay Biswas,² and Chinmay Biswas³

¹Department of Mathematics, University of Kalyani, Kalyani, Nadia, West Bengal 741235, India

²Rajbari, Rabindrapalli, R. N. Tagore Road, P.O. Krishnagar, Nadia,West Bengal 741101, India

³Taraknagar Jamuna Sundari High School, Vill+P.O. Taraknagar, P.S. Hanskhali, Nadia, West Bengal 741502, India

Received 1 May 2014; Accepted 15 June 2014; Published 17 August 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 Sanjib Kumar Datta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove some results relating to the comparative growth properties of iterated entire functions using (p, q)th order ((p, q)th lower order).

1. Introduction, Definitions, and Notations

Let be an entire function defined in the open complex plane . The maximum term , the maximum modulus and Nevanlinna’s characteristic function of on are, respectively, defined as , and where for all . We do not explain the standard definitions and notations in the theory of entire function as those are available in [1]. In the sequel the following two notations are used:

To start our paper we just recall the following definitions.

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

Definition 2 (see [2]). Let be an integer . The generalised order and generalised lower order of an entire function are defined as

When , Definition 2 coincides with Definition 1.

Definition 3. A function is called a generalised proximate order of a meromorphic function relative to if

(i) is nonnegative and continuous for ,(ii) is differentiable for except possibly at isolated points at which and exist,(iii), (iv),(v).

The existence of such a proximate order is proved by Lahiri [3].

Similarly one can define the generalised lower proximate order of in the following way.

Definition 4. A function is defined as a generalised lower proximate order of a meromorphic function relative to if

(i) is nonnegative and continuous for ,(ii) is differentiable for except possibly at isolated points at which and exist,(iii), (iv),(v).

Definitions 3 and 4 are both valid for entire .

Juneja et al. [4] defined the th order and th lower order of an entire function , respectively, as follows:

where are positive integers with .

For and , we respectively denote and by and .

Since for , {cf. [5]}

It is easy to see that

According to Lahiri and Banerjee [6] if and are entire functions, then the iteration of with respect to is defined as follows:

, according to the fact that    is odd or even, and so

Clearly all and are entire functions.

In this paper we would like to investigate some growth properties of iterated entire functions on the basis of their maximum terms, th order and th lower order where ,   are positive integers with .

2. Lemmas

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

Lemma 5 (see [7]). If and are any two entire functions then for all sufficiently large values of ,

Lemma 6. Let and be any two entire functions such that and where ,  ,  , and are any four positive integers with and . Then for any even number and for all sufficiently large values of ,

and for any odd number (≠1) and for all sufficiently large values of ,

where is any arbitary number.

Proof. Let us consider to be an even number.

Then in view of Lemma 5 and the inequality   {cf. [5]} we get for all sufficiently large values of that

Case I. Let and . We obtain from (12) for all sufficiently large values of that

Therefore,

Similarly,

This establishes the first and the tenth part of the lemma, respectively.

Case II. Let and . Now we get from (13) for all sufficiently large values of that

Thus,

Analogously,

from which the second and the eleventh part of the lemma follow.

Case III. Let and . Therefore it follows from (12) for all sufficiently large values of that

Therefore,

Similarly,

This proves the third and the twelfth part of the lemma, respectively.

Case IV. Let . Now we have from (12) for all sufficiently large values of that

Thus,

Analogously,

This establishes the fourth and the thirteenth part of the lemma, respectively.

Case V. Let . We obtain from (12) for all sufficiently large values of that

Therefore,

Similarly,

from which the fifth and the fourteenth part of the lemma follow.

Case VI. Let , , . We get from (13) for all sufficiently large values of that

Thus,

Analogously,

and this proves the sixth and the fifteenth part of the lemma, respectively.

Case VII. Let  , , . Therefore it follows from (28) for all sufficiently large values of that

Therefore,

Similarly,

This establishes the seventh and the sixteenth part of the lemma respectively.

Case VIII. Let ,  ,  . We obtain from (12) for all sufficiently large values of that

Thus,

Analogously,

from which the eighth and the seventeenth part of the lemma follow.

Case IX. Let ,  ,  . We get from (34) for all sufficiently large values of that

Therefore,

Similarly,

This proves the ninth and the eighteenth part of the lemma, respectively.

Thus the lemma follows.

Lemma 7 (see [8]). Let be an entire function. Then for any the function is an increasing function of .

Lemma 8 (see [8]). Let be an entire function. Then for any the function is an increasing function of .

3. Theorems

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

Theorem 9. If and are any two entire functions such that and are both finite where are positive integers with and , then for any even number and ,

Proof. Putting in the inequality   .  [2]} and in view of the inequality we get that

Let .

Since , for given we get for a sequence of values of tending to infinity that

and for all large positive numbers of ,

Since , for a sequence of values of tending to infinity we get for any that

because is an increasing function of by Lemma 7.

Since and are both arbitrary, we get from above that

Again let .

Since , in view of condition (v) of Definition 4 it follows for a sequence of values of tending to infinity and for given that

and for all sufficiently large values of ,

As , for a sequence of values of tending to infinity we get for any that

because is an increasing function of by Lemma 7.

Since and are both arbitrary, we get from (49) that

Case I. Let and . Then from the third part of Lemma 6, we obtain for all sufficiently large values of that

Since we get from (42) and above that

Case II. Let ,   and . Then from the eighth part of Lemma 6, we obtain for all sufficiently large values of that

Since we get from (42) and above that

Case III. Let ,   and . Then from the ninth part of Lemma 6, we obtain for all sufficiently large values of that

Since we get from (42) and above that

Case IV. Let and . Then from the first part of Lemma 6 and (42), we get for all sufficiently large values of that

Therefore we get from above that

Case V. Let , and . Then from the second part of Lemma 6 and (42) we get for all sufficiently large values of that

Therefore we get from above that

Case VI. Let and . Then from (42) and the fifth part of Lemma 6, we get for all sufficiently large values of that

Therefore we get from above that

Now from (52) of Case I and (46), it follows that

This proves the first part of the theorem.

Again for , we obtain in view of (50) and (52) of Case I that

Thus the second part of the theorem follows.

Again from (54) of Case II and (46), it follows that

This proves the third part of the theorem.

Similarly, from (56) of Case III and (46), it follows that

This proves the fourth part of the theorem.

Again for , we obtain in view of (50) and (56) of Case III that

Thus the fifth part of the theorem follows.

Also from (58) of Case IV and (46), it follows that

Hence the sixth part of the theorem is established.

Similarly, from (60) of Case V and (46), it follows that

This concludes the seventh part of the theorem.

Also from (62) of Case VI and (46), we obtain that

Thus the eighth part of the theorem is proved.

Corollary 10. Under the same conditions of Theorem 9, if then

Proof. In view of (41) and from (49) we have for a sequence of values of tending to infinity that

Case I. Let and . Then from the third part of Lemma 6 we obtain for all sufficiently large values of that

Now combining (72) and (73) it follows for a sequence of values of tending to infinity that

So from above we obtain that

Thus the first part of the corollary follows.

Case II. Let , and . Then from the ninth part of Lemma 6 we obtain for all sufficiently large values of that

Therefore combining (72) and (76) it follows for a sequence of values of tending to infinity that

So from above we obtain that

Thus the second part of the corollary follows.

In the line of Theorem 9, we may state the following theorem without its proof.

Theorem 11. Let and be any two entire functions such that and are both finite where ,  ,   are positive integers with and . Then for any ,

when is odd and .

Corollary 12. Under the same conditions of Theorem 11, if then

The proof of the Corollary 12 is omitted as it can be carried out in the line of Theorem 11 and Corollary 10.

Theorem 13. Let and be any two entire functions such that and are both finite where ,  ,   are positive integers with and . Then for any even number ,

Proof.  

Case I. Let . As , for given we obtain for all sufficiently large values of that

and for a sequence of values of tending to infinity,

Since , for a sequence of values of tending to infinity we get for any that

because is an increasing function of by Lemma 8.

Since and are both arbitrary, we get from above that

Case II. Let . Since , in view of condition (v) of Definition 3 it follows for all sufficiently large values of and for a given that

and for a sequence values of tending to infinity

As , for a sequence of values of tending to infinity we get for any that

because is an increasing function of by Lemma 8.

Since and are both arbitrary, we get from above that

Therefore from (52) and (85) it follows that

This proves the first part of the theorem.

Again for , we obtain in view of (52) and (89) that

Thus the second part of the theorem is established.

Similarly, from (58) and (85) we get that

Thus the seventh part of the theorem follows.

Now, in the line of Theorem 9 and using the similar manner, one may easily prove the remaining parts of Theorem 13.

Theorem 14. Let and be any two entire functions such that and are both finite where ,  ,   are positive integers with and . Then

when is odd and .

The proof of Theorem 14 is omitted as it can be carried out in the line of Theorem 13.

Theorem 15. Let and be any two entire functions such that where ,  ,  ,   are positive integers with and . Then

where is any even number.

Proof. Since we can choose in such a way that

Now for all sufficiently large values of ,

Again for all sufficiently large values of , we obtain that

Now the following cases may arise.

Case I. Let and . Then we have, from the third part of Lemma 6, for all sufficiently large values of that

Now from (97) and (100) we have for all sufficiently large values of that

Case II. Let . Now we obtain from the fourth part of Lemma 6 for all sufficiently large values of that

Then from (97) and (102), we obtain for all sufficiently large values of that

Case III. Let ,   and . Then we get from Lemma 6 for all sufficiently large values of that

Now from (97) and (104), we get for all sufficiently large values of that

Case IV. Let , and . Now we obtain from Lemma 6 for all sufficiently large values of that

Then from (97) and (106), we have for all sufficiently large values of that

Further from (96), it follows for all sufficiently large values of that

Case V. Let and . Then for all sufficiently large values of we get in view of the first part of Lemma 6 that

Now from (108) and (109), we have for all sufficiently large values of that

Case VI. Let , and . Now for all sufficiently large values of we get in view of the second part of Lemma 6 that

Then from (108) and (111), we have for all sufficiently large values of that

Case VII. Let . Then for all sufficiently large values of we get in view of the fifth part of Lemma 6 that

Now from (108) and (113), we have for all sufficiently large values of that

Case VIII. Let , and . Now for all sufficiently large values of we get in view of the sixth part of Lemma 6 that

Then from (108) and (104), we have for all sufficiently large values of that

Case IX. Let ,   and . Then for all sufficiently large values of we get in view of the seventh part of Lemma 6 that

Now from (108) and (117), we have for all sufficiently large values of that

Now combining (101) of Case I and (99), we get for all sufficiently large values of that

Now in view of (95), it follows from (119) that

This proves the first part of the theorem.

Similarly combining (95), (103) of Case II, and (99), we obtain for all sufficiently large values of that

Further combining (105) of Case III and (99) and in view of (95) it follows for all sufficiently large values of that

Thus the second part of the theorem follows from (121) and (122).

Again combining (107) of Case IV and (99), we get for all sufficiently large values of that

Therefore in view of (95) we get from (123) that

This proves the third part of the theorem.

Similarly combining (110) of Case V and (99), we obtain in view of (95) for all sufficiently large values of that

This establishes the fourth part of the theorem.

Analogously, in view of (95), (112) of Case VI, and (99) it follows for all sufficiently large values of that

Thus the fifth part of the theorem follows from above.

Again combining (95), (114) of Case VII, and (99) we obtain for all sufficiently large values of that

which is the sixth part of the theorem.

Further in view of (95), (116) of Case VIII, and (99) we get for all sufficiently large values of that

This proves the seventh part of the theorem.

Similarly, combining (95), (118) of Case IX, and (99) it follows for all sufficiently large values of that

This establishes the eighth part of the theorem.

Remark 16. The condition in Theorem 15 is essential as we see in the following example.

Example 17. Let and , .

Then.

Now

Then

which is contrary to Theorem 15.

Remark 18. Theorem 15 is still valid with “limit inferior” instead of “limit” if we replace the condition “ ” by “ .”

Remark 19. Considering and ,  , one can easily verify that the condition in Remark 18 is essential.

In the line of Theorem 15, we may state the following theorem without its proof.

Theorem 20. Let and be any two entire functions such that where ,  ,  ,   are positive integers with and . Then

when is odd and .

The proof is omitted.

Remark 21. In Theorem 20 if we take the condition instead of , then also Theorem 20 remains true with “limit inferior” in place of “ limit.”

Theorem 22. Let and be any two entire functions such that and where ,  ,  ,   are positive integers with and . Then

where is any even number.

Proof. Let and .

Now from (101) we get for all sufficiently large values of that

Therefore combining (98) and (136), we get for all sufficiently large values of that

This proves the first part of the theorem.

Further suppose that . Then from (98) and (103), we obtain for all sufficiently large values of that

Again let ,   and . Then also combining (98) and (105), it follows for all sufficiently large values of that

Thus the second part of the theorem follows from (138) and (139).

Similarly, using the same technique of above one can easily prove the remaining parts of Theorem 22 from (107), (110), (112), (114), (116), and (118), respectively, and with the help of inequality (98). Hence their proofs are omitted.

Remark 23. The condition in Theorem 22 is necessary which is evident from the following example.

Example 24. Let , and ,  .

Then

Now

Therefore

which is contrary to Theorem 22.

Remark 25. Theorem 22 is still valid with “limit inferior” instead of “limit superior” if we replace by .

Remark 26. Considering , and , , one can easily verify that the condition in Remark 25 is essential.

Theorem 27. Let and be any two entire functions such that and where ,  ,  ,   are positive integers with and . Then

when is odd and .

The proof is omitted.

Remark 28. In Theorem 27 if we replace by and the other conditions remaining the same, then also Theorem 20 remains true with “limit inferior” in place of “limit superior.”

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.

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