International Journal of Analysis

Volume 2013, Article ID 538027, 12 pages

http://dx.doi.org/10.1155/2013/538027

## Uniqueness of Meromorphic Functions Sharing Fixed Point

Department of Mathematics, Karnatak University, Dharwad 580003, India

Received 28 September 2012; Revised 22 January 2013; Accepted 30 January 2013

Academic Editor: Alexandre Timonov

Copyright © 2013 Subhas S. Bhoosnurmath and Veena L. Pujari. 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 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,