Journal of Complex AnalysisVolume 2013 (2013), Article ID 395067, 9 pageshttp://dx.doi.org/10.1155/2013/395067

Research Article

Growth Analysis of Wronskians in terms of Slowly Changing Functions

Sanjib Kumar Datta,¹ Tanmay Biswas,² and Sultan Ali³

¹Department of Mathematics, University of Kalyani, Kalyani, Nadia, West Bengal 741235, India²Rajbari, Rabindrapalli, RN. Tagore Road, Krishnagar, Nadia, West Bengal 741101, India³Department of Mathematics, Kalna College, Kalna, Burdwan, West Bengal 713409, India

Received 12 August 2012; Accepted 29 September 2012

Academic Editor: Vladislav Kravchenko

Copyright © 2013 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.

In the paper we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using generalised -order and generalised -type and Wronskians generated by one of the factors.

1. Introduction, Definitions, and Notations

We denote by the set of all finite complex numbers. Let be a meromorphic function defined on . We use the standard notations and definitions in the theory of entire and meromorphic functions which are available in [1] and [2]. In the sequel we use the following notation: for . and .

The following definitions are well known.

Definition 1. A meromorphic function is called small with respect to if .

Definition 2. Let be linearly independent meromorphic functions and small with respect to . We denote by the Wronskian determinant of , that is,

Definition 3. If , the quantity is called the Nevanlinna deficiency of the value “.”

From the second fundamental theorem it follows that the set of values of , the quaintity for which is countable and (cf. [1, ]). If, in particular, , we say that has the maximum deficiency sum.

Let be a positive continuous function increasing slowly, that is, as for every positive constant . Singh and Barker [3] defined it in the following way.

Definition 4 (see [3]). A positive continuous function is called a slowly changing function if, for , uniformly for . If further, is differentiable, the above condition is equivalent to

Somasundaram and Thamizharasi [4] introduced the notions of -order and -type for entire functions. The more generalised concept for -order and -type for entire and meromorphic functions are -order and -type, respectively. In the line of Somasundaram and Thamizharasi [4], for any positive integer one may define the generalised -order (generalised -lower order and generalised -type in the following manner.

Definition 5. The generalised -order and the generalised -lower order of an entire function are defined as When is meromorphic, it can be easily verified that

Definition 6. The generalised -type of an entire function is defined as follows: For meromorphic ,

For , we may get the classical cases {cf. [4]} of Definitions 5 and 6, respectively.

Lakshminarasimhan [5] introduced the idea of the functions of L-bounded index. Later Lahiri and Bhattacharjee [6] worked on the entire functions of L-bounded index and of nonuniform L-bounded index. Since the natural extension of a derivative is a differential polynomial, in this paper we prove our results for a special type of linear differential polynomials, namely, the Wronskians. In the paper we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using generalised -order and generalised -type and wronskians generated by one of the factors.

2. Lemmas

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

Lemma 7 (see [7]). If   be meromorphic and   be entire then for all sufficiently large values of ,

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

Lemma 9 (see [9]). Let be a transcendental meromorphic function having the maximum deficiency sum. Then

Lemma 10 (see [10]). If be a transcendental meromorphic function having the maximum deficiency sum. Then the generalised -order (generalised -lower order) of and that of are the same.

Lemma 11. Let be a transcendental meromorphic function having the maximum deficiency sum. Then

Proof. By Lemmas 9 and 10 we get that Also by Lemma 9,   exists and is equal to for . Therefore Thus the lemma follows.

3. Theorems

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

Theorem 12. Let be a transcendental meromorphic function having the maximum deficiency sum and entire such that where is any positive integer. Then for any positive real number , where and

Proof. Let . Using the definition of generalised -lower order we obtain in view of Lemma 8 for a sequence of values of tending to infinity that Again in view of Lemma 10 we have for all sufficiently large values of that Now from (17) and (18) it follows for a sequence of values of tending to infinity that Again from (19) we get for a sequence of values of tending to infinity that Case I. If then it follows from (20) that Case II. then the following two subcases may arise.Subcase (a). If  , then we get from (21) that Subcase (b). If    then and we obtain from (21) that Combining Cases I and II we obtain that where This proves the theorem.

Theorem 13. Let be a meromorphic function and transcendental entire such that , and where . Then for any positive real number , where and

The proof is omitted because it can be carried out in the line of Theorem 12.

Theorem 14. Let be a transcendental meromorphic function with and entire such that and where is any positive integer. If as and for some positive , then

Proof. Since and by Lemma 7 we get for all sufficiently large values of that As , we can choose in such a way that and since as , we obtain that Now from (32) and (33) and in view of Lemma 10 it follows that Thus the theorem is established.

In the line of Theorem 14 the following theorem can be proved.

Theorem 15. Let be meromorphic and transcendental entire having the maximum deficiency sum such that , and where . If as and for some positive , then The proof is omitted.

Theorem 16. Let be a transcendental meromorphic function such that with and . Also let be entire. If for 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 that for all sufficiently large values of and in view of Lemma 10, Thus from (37) and (38) we have for a sequence of values of tending to infinity that This is a contradiction. This proves the theorem.

Remark 17. Theorem 16 is also valid with “limit superior" instead of “limit" if is replaced by and the other conditions remain the same.

Corollary 18. Under the assumptions of Theorem 16 or Remark 17,

Proof. From Theorem 16 or Remark 17 we obtain for all sufficiently large values of and for , from which the corollary follows.

Remark 19. The condition in Theorem 16 and Corollary 18 is necessary which is evident from the following example.

Example 20. Let , , and where is any positive real number. Then Now taking and we get from Definition 2 that Now Therefore which is a contradiction.

Remark 21. Considering , , , , for any positive real number and taking , in Definition 2, one can easily verify that the condition in Remark 17 and Corollary 18 is essential.

Theorem 22. Let be meromorphic and transcendental entire such that with and where is any positive integer. Also let . Then

We omit the proof of Theorem 22 because it can be carried out in the line of Theorem 16.

Remark 23. Theorem 22 is also valid with “limit superior" instead of “limit" if is replaced by and the other conditions remain the same.

In the line of Corollary 18 we may easily verify the following.

Corollary 24. Under the assumptions of Theorem 22 or Remark 19,

Remark 25. Taking ,, , , for any positive real number and choosing , in Definition 2 we may establish the necessity of the conditions and , respectively, in Theorem 22, Remark 23, and Corollary 24.

Theorem 26. Let be meromorphic function and entire function having the maximum deficiency sum such that and where is any positive integer. Then if then and if then

Proof. Since in view of Lemma 7 we obtain for all sufficiently large values of that Using the definition of -type we obtain from (50) for all sufficiently large values of that Again from the definition of -type and in view of Lemmas 10 and 11 we get for a sequence of values of tending to infinity that Now from (51) and (52) it follows for a sequence of values of tending to infinity that If then from (53) we get that Since is arbitrary it follows from above that Thus the first part of Theorem 26 follows. Again if then from (53) it follows that As is arbitrary we obtain from the above that Thus the second part of Theorem 26 follows.

Theorem 27. Let be a transcendental meromorphic function having the maximum deficiency sum and entire such that , , and where . Then if as and for some positive then and if then

Proof. In view of condition we obtain from (51) for all sufficiently large values of that Again from the definition of -type and in view of Lemma 10 we get for a sequence of values of tending to infinity that Now from (60) and (61) it follows for a sequence of values of tending to infinity that If then from (62) we get that Since is arbitrary, it follows from the above that Now in view of Lemma 11 we get from the above that Thus the first part of Theorem 27 follows. Again if then from (62) it follows that As is arbitrary, we obtain from the above that Thus the second part of Theorem 27 follows.

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