Abstract
We study the uniqueness of meromorphic functions concerning differential polynomials sharing fixed point and obtain some significant results , which improve the results due to Lin and Yi (2004).
1. Introduction and Main Results
Let be a nonconstant meromorphic function in the whole complex plane . We will use the following standard notations of value distribution theory: , (see [1, 2]). We denote by any function satisfying possibly outside of a set with finite measure.
Let be a finite complex number and a positive integer. We denote by the counting function for the zeros of in with multiplicity and by the corresponding one for which multiplicity is not counted. Let be the counting function for the zeros of in with multiplicity and the corresponding one for which multiplicity is not counted. Set
Let be a nonconstant meromorphic function. We denote by the counting function for -points of both and about which has larger multiplicity than , where multiplicity is not counted. Similarly, we have notation .
We say that and share CM (counting multiplicity) if and have same zeros with the same multiplicities. Similarly, we say that and share IM (ignoring multiplicity) if and have same zeros with ignoring multiplicities.
In 2004, Lin and Yi [3] obtained the following results.
Theorem A. Let and be two transcendental meromorphic functions, an integer. If and share CM, then either or where is a nonconstant meromorphic function.
Theorem B. Let and be two transcendental meromorphic functions, an integer. If and share CM, then .
In this paper, we study the uniqueness problems of entire or meromorphic functions concerning differential polynomials sharing fixed point, which improves Theorems A and B.
1.1. Main Results
Theorem 1. Let and be two nonconstant meromorphic functions, a positive integer. If and share CM, and share IM, then either or where is a nonconstant meromorphic function.
Theorem 2. Let and be two nonconstant meromorphic functions, a positive integer. If and share CM, and share IM, then .
Theorem 3. Let and be two nonconstant entire functions, an integer. If and share CM, then .
2. Some Lemmas
Lemma 4 (see [4]). Let , , and be nonconstant meromorphic functions such that . If , , and are linearly independent, then where and .
Lemma 5 (see [1]). Let and be two nonconstant meromorphic functions. If , where , , and are non-zero constants, then Lemmas 4 and 5 play a very important role in proving our theorems.
Lemma 6 (see [1]). Let be a nonconstant meromorphic function and let be a nonnegative integer, then The following lemmas play a cardinal role in proving our results.
Lemma 7. Let and be nonconstant meromorphic functions. If and share CM and , then
Proof. Applying Nevanlinna’s second fundamental theorem to , we have By the first fundamental theorem and (9), we have We know that Therefore using Lemma 6, (10) becomes since , we have This completes the proof of Lemma 7.
Lemma 8. Let and be nonconstant entire functions. If and share CM and , then
Proof. Applying Nevanlinna’s second fundamental theorem to , we have Since is an entire function, we have and the above equation becomes By the first fundamental theorem and (17), we have We know that Therefore using Lemma 6, (18) becomes or since , we have This completes the proof of Lemma 8.
Lemma 9 (see [5]). Suppose that is a meromorphic function in the complex plane and , where , are small meromorphic functions of . Then
Lemma 10 (see [6]). Let , and be three meromorphic functions satisfying , let , , and . If , and are linearly independent, then , and are linearly independent.
3. Proof of Theorems
Proof of Theorem 1. By assumption, and share CM, and and share IM. Let
Then, is a meromorphic function satisfying
Therefore,
From (24), we easily see that the zeros and poles of are multiple and satisfy
Let
Then, and denote the maximum of , .
We have
Therefore, , and thus
Now, we discuss the following three cases.
Case 1. Suppose that neither nor is a constant.
If , , and are linearly independent, then by Lemma 4 and (28), we have
Using (27), we note that
Since , we obtain that
But , so we get
Using (33) and (35) in (31), we get
Since and share IM, we have .
Using this with (27), we get
If is a zero of with multiplicity , then is a zero of with multiplicity ; we have
Similarly,
Let
By Lemma 9, we have .
Since , we have
By the first fundamental theorem, we have
we have
From (37)–(43), we get
Using Lemma 6, we get
Let
Then . By Lemma 10, , and are linearly independent. In the same manner as above, we get
Note that
Adding (45) and (47) gives
Using (48), we get
or
Combining (50) and (51), we get
By and (30), we get a contradiction. Thus , , and are linearly dependent. Then, there exists three constants such that
If , from (53) , , and
On integrating, we get
since , we get a contradiction.
Thus , and by (53) we have
Substituting this in , we get
that is,
From (28), we obtain
Applying Lemma 5 to the above equation, we get
Note that
Using (61), we get
By Lemmas 9 and 6 and (63), we have
we obtain , which contradicts .
Case 2. Suppose that , where is a constant.
If , then we have
Applying Lemma 5 to the above equation, we have
Note that
Therefore,
Using Lemmas 9 and 6 and (69), we have
Using Lemma 7, we get
since , we get a contradiction.
Therefore , and by (27) and (24) we have
On integrating, we get
We claim that . Suppose that , then
We have
Similarly,
Using Lemma 9, we have
Thus,
Similarly,
Therefore, (74) becomes
which contradicts . Thus, we have
Let . If , then we easily obtain that
If , that is, .
Case 3. Suppose that , where is a constant.
If , then we have
Applying Lemma 5 to the above equation, we have
Note that
Therefore using (84), we have
Using Lemmas 9 and 6 and (86), we have
Using Lemma 7, we get
since , we get a contradiction. Thus, . Hence,
Let be a zero of of order . From (89), we know that is a pole of . Suppose that is a pole of of order . From (89), we obtain
which implies that and . Hence,
Let be a zero of of order , then from (89) is a pole of (say order ). By (89), we get
Let be a zero of of order that is not zero of , then from (89), is a pole of of order . Again by (89), we get
In the same manner as above, we have similar results for the zeros of . From (89)–(93), we have
that is,
By Nevanlinna’s second fundamental theorem, we have from (91), (92), and (95) that
Similarly,
From (96) and (97), we get
since , we get a contradiction.
This completes the proof of Theorem 1.
Using the same argument as in the proof of Theorem 1, we can prove Theorem 2.
Proof of Theorem 3. By the assumption of the theorem, we know that either both and are two transcendental entire functions or both and are polynomials.
If and are transcendental entire functions, putting , and using similar arguments as in the proof of Theorem 1, we easily obtain Theorem 3.
If and are polynomials, and share CM, we get
where is a nonzero constant. Suppose that , (99) can be written as
Applying Lemma 5 to the above equation, we have
Since is a polynomial, so it does not have any poles. Thus, we have,
Note that
Therefore,
Using Lemmas 9 and 6 and (104), we have
Using Lemma 8, we get
since , we get a contradiction.
Therefore, ; so (99) becomes
On integrating, we get
We claim that . Suppose that , then
We have
Similarly,
Using Lemma 9, we have
Thus,
Similarly,
Therefore, (109) becomes
which contradicts . Thus, we have
Let . If , then by (116) we have
By Picard’s theorem, is a constant. Hence, is a constant, which is a contradiction. Therefore, , that is, .
4. Remarks
If the condition “ and share CM” is replaced by the condition “ and share CM,” where is a meromorphic function such that and ; the conclusion of Theorems 1, 2, and 3 still holds. We, thus, obtain the following results.
Theorem 11. Let and be two nonconstant meromorphic functions, a positive integer. If and share CM, and share IM, then either or where is a nonconstant meromorphic function.
Theorem 12. Let and be two nonconstant meromorphic functions, a positive integer. If and share CM, and share IM, then .
Theorem 13. Let and be two nonconstant entire functions, an integer. If and share CM, then .
Acknowledgments
The authors thank the referees for their 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 (no. SR/S4/MS: 520/08).