Abstract

We study the uniqueness of meromorphic functions and differential polynomials sharing one value with weight and prove two main theorems which generalize and improve some results earlier given by M. L. Fang, S. S. Bhoosnurmath and R. S. Dyavanal, and so forth.

1. Introduction and Results

Let be a nonconstant meromorphic function defined in the whole complex plane . It is assumed that the reader is familiar with the notations of the Nevanlinna theory such as , , , and , that can be found, for instance, in [1–3].

Let and be two nonconstant meromorphic functions. Let be a finite complex number. We say that and share the value CM (counting multiplicities) if and have the same zeros with the same multiplicities, and we say that and share the value IM (ignoring multiplicities) if we do not consider the multiplicities. When and share 1 IM, let be a 1-point of of order and a 1-points of of order ; we denote by the counting function of those 1-points of and , where and by the counting function of those 1-points of and , where . is the counting function of those 1-points of both and , where . In the same way, we can define , , and . If and share 1 IM, it is easy to see that

Let be a nonconstant meromorphic function. Let be a finite complex number and a positive integer; we denote by (or ) the counting function for zeros of with multiplicity (ignoring multiplicities) and by (or ) the counting function for zeros of with multiplicity at least (ignoring multiplicities). Set We further define

In 2002, C. Y. Fang and M. L. Fang [4] proved the following result.

Theorem A (see [4]). Let and be two nonconstant entire functions, and let (≥8) be a positive integer. If and share 1 CM, then .

Fang [5] proved the following result.

Theorem B (see [5]). Let and be two nonconstant entire functions, and let , be two positive integers with . If and share 1 CM, then .

In [6], for some general differential polynomials such as , Liu proved the following result.

Theorem C (see [6]). Let and be two nonconstant entire functions, and let be three positive integers such that . If and share 1 IM, then either or and satisfy the algebraic equation , where .

The following example shows that Theorem A is not valid when and are two meromorphic functions.

Example 1.1. Let , , where . Then and share 1 CM, but .

Lin and Yi [7] and Bhoosnurmath and Dyavanal [8] generalized the above results and obtained the following results.

Theorem D (see [7]). Let and be two nonconstant meromorphic functions with , and let be a positive integer. If and share 1 CM, then .

Theorem E (see [8]). Let and be two nonconstant meromorphic functions satisfying , and let be two positive integers with . If and share 1 CM, then .

Naturally, one may ask the following question: is it really possible to relax in any way the nature of sharing 1 in the above results?

To state the next result, we require the following definition.

Definition 1.2 (see [9]). Let be a nonnegative integer or infinity. For one denotes by the set of all -points of , where an -point of multiplicity is counted times if and times if . If , one says that share the value with weight .
We write that share to mean that share the value with weight ; clearly if share , then share for all integers with . Also, we note that share a value   IM or CM if and only if they share or , respectively.

Recently, with the notion of weighted sharing of values, Xu et al. [10] improved the above results and proved the following theorem.

Theorem F (see [10]). Let and be two nonconstant meromorphic functions, and let be two positive integers with . If , , and share 1 , then .

Theorem G (see [10]). Let and be two nonconstant meromorphic functions, and let be two positive integers with . If , , and share 1 , then .

Remark 1.3. The proof of Theorem E contains some mistakes: for example, one cannot get formulas (6.9) and (6.10) in [8]. Therefore, the last inequality in page 1203 of [8] does not hold. So, Theorem E will not stand. Similarly, in [10], the proof of Case  1 in Theorem F is incorrect (see page 63 of paper [10]). Hence, the conclusion of Theorems F and G will not stand.
Now one may ask the following questions which are the motivations of the paper.(i)What happens if the conclusions of Theorems E, F, and G can not stand?(ii)Can the value be further reduced in the above results?
In the paper, we investigate the solutions of the above questions. We improve and generalize the above related results by proving the following theorems.

Theorem 1.4. Let and be two nonconstant meromorphic functions, and let be two positive integers with . If , , and share 1 , then or .

Theorem 1.5. Let and be two nonconstant meromorphic functions, and let be two positive integers with . If , and share 1 , then or .

2. Some Lemmas

For the proof of our result, we need the following lemmas.

Lemma 2.1 (see [1]). Let be nonconstant meromorphic function, and let be finite complex numbers such that . Then

Lemma 2.2 (see [1]). Let be a nonconstant meromorphic function, a positive integer, and a nonzero finite complex number. Then Here is the counting function which only counts those points such that but .

Lemma 2.3 (see [11]). Let be a nonconstant meromorphic function, and let be two positive integers. Then Clearly .

Lemma 2.4 (see [1]). Let be a transcendental meromorphic function, and let , be two meromorphic functions such that , . Then

Lemma 2.5. Let and be two nonconstant meromorphic functions, and let (≥1), (≥1) be two positive integers. Suppose that and share .(i)If and then either or .(ii)If and then either or .

Proof. Let Suppose that .
If is a common simple 1-point of and , substituting their Taylor series at into (2.7), we see that is a zero of . Thus, we have
By our assumptions, have poles only at zeros of and and poles of and , and those 1-points of and whose multiplicities are distinct from the multiplicities of correspond to 1-points of and , respectively. Thus, we deduce from (2.7) that here has the same meaning as in Lemma 2.2.
By Lemma 2.2, we have Since and share , we get We obtain from (2.8)–(2.11) that (i)If , it is easy to see that Since combining with (2.12), (2.13), and (2.14), we obtain Without loss of generality, we suppose that there exists a set with infinite measure such that for : for and , that is, that is, , which contradicts hypothesis (2.5).
Therefore, we have , that is, By solving this equation, we obtain where are two constants. Next, we consider three cases.
Case 1 (). For more details see the following subcases.Subcase 1.1 (). Then, by (2.19), we have .
By Lemma 2.2 and Lemma 2.3, we get that is, . Thus, by (2.5) we deduce that for , a contradiction.Subcase 1.2 (). Then, by (2.19), we have .
Therefore .
By Lemma 2.2 and Lemma 2.3, we get that is, . Thus, by (2.5) we deduce that for , a contradiction.Case 2 (). For more details see the following subcases.Subcase 2.1 (). Then by (2.19), we get . So, we have . We can deduce a contradiction as in Case 1.Subcase 2.2 (). Then by (2.19), we get .Case 3 (). For more details see the following subcases.Subcase 3.1 (). Then by (2.19), we get . So, we have . We can deduce a contradiction as in Case 1.Subcase 3.2 (). Then by (2.19), we get . From this, we obtain , where is a polynomial, and so . If , then, by Lemma 2.4, we get Thus, by (2.5) we deduce that for , a contradiction. Therefore, we conclude that , that is, .
(ii) If , it is easy to see that By Lemma 2.3, we can get Combining with (2.12), (2.24), and (2.14), we obtain
Without loss of generality, we suppose that there exists a set with infinite measure such that for : for and , that is, that is, 9, which contradicts hypothesis (2.6).
Therefore, we have , that is, By solving this equation, we obtain where are two constants. Using the argument in (i), we can obtain or . We here omit the details.
The proof of Lemma 2.5 is completed.

3. Proof of Theorem 1.4

Proof. Let and . We have Since similarly, Next, by Lemma 2.1, we have Similarly From (3.2)–(3.5), we get Since , we get . Considering that and share , then, by Lemma 2.5, we deduce that either or .
Next, we consider the following two cases.Case 1. , that is, .Case 2. , that is, Suppose that ; then we consider following two cases.(i)Let be a constant. Then from (3.7) it follows that and , which leads to contradiction. (ii)Let be not a constant. Since , we have , and hence we deduce that and , where is a nonconstant meromorphic function. It follows that On the other hand, by the second fundamental theorem, we get where are distinct roots of the algebraic equation .
So we have which contradicts the assumption that . Thus . This completes the proof of Theorem 1.4.

4. Proof of Theorem 1.5

Proof. From (3.2)–(3.5), we get Since , we get . Considering that and share , then, by Lemma 2.5, we deduce that either or .
Next, by using the argument in Theorem 1.4, we obtain the conclusion of Theorem 1.5. We here omit the details.

Acknowledgments

This research was supported by the NSF of Key Lab of Geomathematics of Sichuan Province of China. The author wishes to thank the referee for his/her valuable comments and suggestions.