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International Journal of Analysis
Volume 2013 (2013), Article ID 926340, 8 pages
http://dx.doi.org/10.1155/2013/926340
Research Article

On Entire and Meromorphic Functions That Share One Small Function with Their Differential Polynomial

Department of Mathematics, Karnatak University, Dharwad 580003, India

Received 24 September 2012; Accepted 22 December 2012

Academic Editor: Mats Ehrnström

Copyright © 2013 Subhas S. Bhoosnurmath and Smita R. Kabbur. 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 study the uniqueness of meromorphic functions that share one small function with more general differential polynomial . As corollaries, we obtain results which answer open questions posed by Yu (2003).

1. Introduction and Main Results

In this paper, a meromorphic functions mean meromorphic in the whole complex plane. We use the standard notations of Nevanlinna theory (see [1]). A meromorphic function is called a small function with respect to if , that is, as possibly outside a set of finite linear measure. If and have the same zeros with same multiplicities (ignoring multiplicities), then we say that and share  CM (IM).

For any constant , we denote by the counting function for zeros of with multiplicity no more than and the corresponding for which multiplicity is not counted. Let be the counting function for zeros of with multiplicity at least and the corresponding for which the multiplicity is not counted.

Let and be two nonconstant meromorphic functions sharing value 1 IM. Let be common one point of and with multiplicity and , respectively. We denote by   the counting (reduced) function of those 1 points of where ; by the counting function of those 1-points of where ; by the counting function of those 1-points of where . In the same way, we can define , and (see [2]).

In 1996, Brück [3] posed the following conjecture.

Conjecture 1. Let be a nonconstant entire function such that the hyper-order of is not a positive integer and . If and share a finite value  CM, then , where is a nonzero constant.

In [3], under an additional hypothesis, Brück proved that the conjecture holds when .

Theorem A. Let be a nonconstant entire function. If and share the value 1 CM and if , then , for some constant .
Many people extended this theorem and obtained many results. In 2003, Yu [4] proved the following theorem.

Theorem B. Let . Let be a nonconstant meromorphic function and a meromorphic function such that , and do not have any common pole and as . If and share the value 0 CM and then .

Theorem C. Let . Let be a nonconstant entire function and be a meromorphic function such that and as . If and share the value 0 CM and then .

In the same paper, the author posed the following questions.

Question 1. Can  CM shared value be replaced by an IM shared value in Theorem C?

Question 2. Is the condition sharp in Theorem C?

Question 3. Is the condition sharp in Theorem B?

In 2004, Liu and Gu [5] applied different method and obtained the following theorem which answers some questions posed in [4].

Theorem D. Let . Let be a nonconstant meromorphic function and a meromorphic function such that and as . If and share the value 0 CM and and do not have any common poles of same multiplicity and then .

Theorem E. Let . Let be a nonconstant entire function and a meromorphic function such that and as . If and share the value 0 CM and then .

Recently, Zhang and Lü [6] considered the problem of meromorphic functions sharing one small function with its th derivative and proved the following theorem.

Theorem F. Let , be integers and a nonconstant meromorphic function. Also let be a small meromorphic function with respect to . If and share the value  IM and or and share the value  CM and then .

Regarding these results, a natural question is what can be said when a nonconstant meromorphic function shares one nonzero small meromorphic function with , where is a differential polynomial in .

Definition 2. Any expression of the type is called differential polynomial in of degree , lower degree , and weight , where are nonnegative integers, are meromorphic functions satisfying and
Further, if , then the differential polynomial is called a homogeneous differential polynomial in of degree .

Correspond to the above question, we obtain the following results, which extend and improve Theorems A–F and give answers to the questions possed by Yu [4] for more general differential polynomial.

Theorem 3. Let be a nonconstant meromorphic function and be a small meromorphic function such that . P[f] be a nonconstant differential polynomial in as defined in (7). If and share the value  IM and then .

Remark 4. Taking , that is, , in (9), we get , which improves (5) and extends the theorem to more general differential polynomial as defined in (7).

Theorem 5. Let be a nonconstant meromorphic function and be a small meromorphic function such that . Let a nonconstant differential polynomial in as defined in (7). If and share the value  CM and then .

Remark 6. Taking , that is, , in (10), we get , which improves (6) and extends the theorem to more general differential polynomial as defined in (7).
Remark 6 gives answer to Question  3 of [4].

Theorem 7. Let be a nonconstant entire function and a small meromorphic function such that . Let be a nonconstant differential polynomial in as defined in (7) If and share the value  IM and then .

Remark 8. Taking , that is, , in (11), we get Remark 8 gives answer to Question  1 of Yu [4].

Theorem 9. Let be a nonconstant entire function and be a small meromorphic function such that . be a nonconstant differential polynomial in as defined in (7). If and share the value  CM and then .

Remark 10. Taking , that is, , in (13), we get , which improves Theorem C and extends the theorem to more general differential polynomial as defined in (7).
Remark 10 gives answer to Question  2 of Yu [4].

Remark 11. By proving Remarks 6, 8, and 10 we have answered Questions  3,  1, and 2 (of [4]), respectively, for the case . Theorems 39 improve and generalize Theorems A–F for more general differential polynomial .

2. Lemma

Lemma 12 (see [7]). Let ba a meromorphic function and be a differential polynomial in . Then where .

Lemma 13 (see [8]). Let be a nonconstant meromorphic function, then

Lemma 14 (see [9]). Let where and are two nonconstant meromorphic functions. If and share 1 IM and , then

Lemma 15. Let be a transcendental meromorphic function. Let be defined as in (7). If , we have

Proof. By the first fundamental theorem, we have We have or By (21), (23) and Lemma 12, we obtain (19).
Since we get Substituting (25) in (19), we obtain (20).

Lemma 16 (see [10]). Let be a transcendental meromorphic function, a differential polynomial in of degree and weight . Then , .

3. Proof of Theorems

Proof of Theorem 3. Let From the conditions of Theorem 3, we know that and share 1 IM. From (26), we have
Let be defined by (17). Suppose that . By Lemma 14, (18) holds.
From (17) and (28), we have where denotes the counting function corresponding to the zeros of which are not the zeros of and . Similarly, is defined.
From the second fundamental theorem, we have Since and share 1 IM, we get from (33):
From this, (18), and (34), we have It is clear that Combining (37), and (38), we obtain Substituting (39) in (35) and using (28), we obtain Using (26) and (19), we get
From (16), (20), and (26) we have From (41) and (42), we get Therefore, we have which is a contradiction to our hypothesis (9).
Thus . By integration, we get from (17) that where and are constants. Thus We discuss the following three cases.
Case 1. Suppose that . From (46), we have From this and second fundamental theorem, we have Therefore, we have which is a contradiction to our hypothesis (9).
Case 2. Suppose that , From (46), we get we claim .
If from (50), we obtain From this, second fundamental theorem, and (20), we have Hence, we have which is a contradiction to our hypothesis (9)
Thus, .
From (50) we have .
Therefore, we have .
Case 3. Suppose that , from (46) we have If , we obtain from (54) that By the same argument as in Case 2, we obtain a contradiction. Hence, .
From (54), we get that is, From (57), we have Using (54), (57), Lemma 12, and first fundamental theorem, we get From this, we have which is a contradiction. This completes the proof of Theorem 3.

Proof of Theorem 5. Let and be given by (26). From the assumption of Theorem 5, we know that and share 1 CM: Proceeding as in Theorem 3, we obtain (41).
Using (61) in (41), we get We have which contradicts (10).
Thus, . Proceeding as in Theorem 3, we prove Theorem 5.

Proof of Theorem 7. is a nonconstant entire function. Taking in proof of Theorem 3, we obtain Theorem 7.

Proof of Theorem 9. is a nonconstant entire function. Taking in proof of Theorem 5, we obtain Theorem 9.

Acknowledgments

The authors thank the referee for his/her valuable suggestions. This research work is supported by the Department of Science and Technology Government of India, Ministry of Science and Technology, Technology Bhavan, New Delhi, India, under the sanction Letter no. (SR/S4/MS: 520/08).

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