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Journal of Mathematics

Volume 2014, Article ID 908798, 18 pages

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

On Iterated Entire Functions with ()th Order

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