On Iterated Entire Functions with (<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.59609pt" id="M1" height="13.8598pt" version="1.1" viewBox="0 -9.26371 26.4627 13.8598" width="26.4627pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.018,0,0,-0.018,10.606,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.018,0,0,-0.018,17.681,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"></path></g></svg>)th Order

Datta, Sanjib Kumar; Biswas, Tanmay; Biswas, Chinmay

doi:https://doi.org/10.1155/2014/908798

Journal of Mathematics

On this page

Abstract References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 908798 | https://doi.org/10.1155/2014/908798

On Iterated Entire Functions with ()th Order

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

Academic Editor: Jen-Chih Yao

Received01 May 2014

Accepted15 Jun 2014

Published17 Aug 2014

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.

References

G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea Publishing, New York, NY, USA, 1949.
D. Sato, “On the rate of growth of entire functions of fast growth,” Bulletin of the American Mathematical Society, vol. 69, pp. 411–414, 1963.
View at: Publisher Site | Google Scholar | MathSciNet
I. Lahiri, “Generalised proximate order of meromorphic functions,” Matematički Vesnik, vol. 41, no. 1, pp. 9–16, 1989.
View at: Google Scholar | MathSciNet
O. P. Juneja, G. P. Kapoor, and S. K. Bajpai, “On the (p,q)-order and lower (p,q)-order of an entire function,” Journal für Die Reine und Angewandte Mathematik, vol. 282, pp. 53–67, 1976.
View at: Google Scholar | MathSciNet
A. P. Singh and M. S. Baloria, “On maximum modulus and maximum term of composition of entire functions,” Indian Journal of Pure and Applied Mathematics, vol. 22, no. 12, pp. 1019–1026, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
B. K. Lahiri and D. Banerjee, “Relative fix points of entire functions,” The Journal of the Indian Academy of Mathematics, vol. 19, no. 1, pp. 87–97, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Clunie, “The composition of entire and meromorphic functions,” in Mathematical Essays Dedicated to A. J. Macintyre, pp. 75–92, Ohio University Press, Athens, Ohio, USA, 1970.
View at: Google Scholar | MathSciNet
S. K. Datta and T. Biswas, “Growth rates of composite entire and meromorphic functions,” Bulletin of the Allahabad Mathematical Society, vol. 27, part 1, pp. 55–94, 2012.
View at: Google Scholar | MathSciNet

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

797

Downloads

875

Citations