Abstract

This note is to show that if is a nonconstant entire function that shares two pairs of small functions ignoring multiplicities with its first derivative , then there exists a close linear relationship between and . This result is a generalization of some results obtained by Rubel and Yang, Mues and Steinmetz, Zheng and Wang, and Qiu. Moreover, examples are provided to show that the conditions in the result are sharp.

1. Introduction and Main Result

Throughout this paper, we use standard notations in the Nevanlinna theory (see, e.g., [1–4]). Let be a meromorphic function. Here and in the following the word “meromorphic” means meromorphic in the whole complex plane. We denote by any real function of growth as outside of a possible exceptional set of finite linear measure. The meromorphic function is called a small function with respect to provided that .

Let and be two nonconstant meromorphic functions, and let and be two small functions with respect to and . If the zeros of and coincide in locations and multiplicities, then we say that and share the pair of small functions CM (counting multiplicities); if we do not consider the multiplicities, then and are said to share the pair of small functions IM (ignoring multiplicities). We see that and share the pair of small functions CM if and only if and share the small function CM, and and share the pair of small functions IM if and only if and share the small function IM. The same argument applies in the case when and are two values in the extended plane.

Moreover, we introduce the following notations. Denote the set of those points by such that is a zero of of multiplicity and a zero of of multiplicity . The set can be similarly defined. Now the notations and denote the counting function and the reduced counting function of with respect to the set , respectively. The notations and can be similarly defined.

Many mathematicians have been interested in the value distribution of different expressions of an entire or meromorphic function and obtained a lot of fruitful and significant results. When dealing with an entire function and its derivative , Rubel and Yang [5] proved the following.

Theorem A. Let be a nonconstant entire function, and let and be distinct finite complex numbers. If and share and CM, then .

Mues and Steinmetz [6] improved Theorem A and obtained the following.

Theorem B. Let be a nonconstant entire function, and let and be distinct finite complex numbers. If and share and IM, then .

When the values and were replaced by two small functions related to , Zheng and Wang [7] proved the following.

Theorem C. Let be a nonconstant entire function, and let and be distinct small functions with respect to . If and share and CM, then .

Recently, Qiu [8] proved the following result which was an improvement of Theorem C.

Theorem D. Let be a nonconstant entire function, and let and be distinct small functions with respect to . If and share and IM, then .

This paper is concerned with what can be said when the IM shared small function is replaced by the IM shared the pair of small functions in Theorem D. In fact, we prove the following result by using the method of [8], which generalizes the above theorems from the point of view of shared pairs.

Theorem 1. Let be a nonconstant entire function, and let , , , and be four small functions of such that none of them is identically equal to and , . If and share and IM, then .

Remark 2. Let and . Then by Theorem 1 we can get Theorem D.

Remark 3. Theorem 1 shows that a nonconstant entire function sharing two pairs of small functions ignoring multiplicities with its first derivative implies that there exists a close linear relationship between them.

Example 4 (see [9]). Let , where Set , . Then and . It is easy to verify that Thus and share and IM, but . This shows that the conclusion in Theorem 1 is not valid generally for a meromorphic function .

Example 5. Let , , , , and . Then and share IM but do not share IM. Clearly, . This shows that the condition in Theorem 1 that and share and IM cannot be weakened.

2. Some Lemmas

Lemma 1. Let be a nonconstant entire function, and let , , , and be four small functions of such that none of them is identically equal to and , . If and share and IM, then .

Proof. Note that and share and IM. By the second fundamental theorem, we get which implies from the definition of that and , respectively.
This completes the proof of Lemma 1.

Lemma 2. Let be a nonconstant entire function, and let , , , and be four small functions of such that none of them is identically equal to and , . If and share and IM, then provided that .

Proof. Note that Since and share and IM, from Lemma 1, (5)–(7), and the condition that is entire, we have
On the other hand, by the second fundamental theorem, Lemma 1, and the condition that is entire, we get
Now (8) and (9) imply
This completes the proof of Lemma 2.

Lemma 3. Let be a nonconstant entire function, and let , , , and be four small functions of such that none of them is identically equal to and , . Suppose that and share and IM. Set If , then(i),(ii) for .

Proof. Since and share and IM, by Lemma 1 we know . Noting and the lemma of the logarithmic derivative, we obtain
Clearly, and . Otherwise from (11) and (12) we have and for nonzero constants , , which implies that and , a contradiction. Then by using a similar method we can deduce that and . It is easy to see by (11) if any zero of of multiplicity is not the pole of and is not the zero of , then it must be a zero of of multiplicity at least. Thus from (6), (7), (11), (13), the condition that and share and IM, and the condition that is entire, we get Likewise, Now by (13) and (17) together with it follows that Thus from this and (19) we have implying (i). Next, it is easy to see that and . For , by (14), (18), , , and the condition that is entire, we have Thus from (20) and (24) it follows that
This proves (ii) and completes the proof of Lemma 3.

Lemma 4 (see [10]; cf. [11, 12]). Let be a nonconstant meromorphic function, and let , where and are differential polynomials in and the degree of is at most . Then

Lemma 5. Let be a nonconstant entire function, and let , , , and be four small functions of such that none of them is identically equal to and , . Suppose that and share and IM. If then .

Proof. Assume that . Let , , , , and be defined by (11)–(15), respectively. Then from the proof process of Lemma 3 we know , , , , , and . Since and share and IM, by Lemmas 1, 2, and 3 it follows that Now from the second fundamental theorem, (27), (28), and the assumption that is entire, we deduce which yields Again by (27), (30), (32), and the assumption that is entire, we obtain
For any , from (13) and (14), we can get .
If , then by (13) and (14) we deduce which implies that where is a nonzero constant. If , then from (35) and the condition that is entire, we obtain , which contradicts (27). If , then we get where is a nonzero constant. We claim that . Indeed, if , then by (36) we deduce which leads to . This contradicts the assumption. Thus and so from (36) we have This and Lemma 4 yield which gives . From this and the condition that is entire, it follows that , a contradiction. Hence , for any positive integers and .
Therefore by (29) and (33) we obtain for any positive integers and . It follows from this, Lemma 1, the second fundamental theorem, and the condition that is entire that which implies that , a contradiction. Thus .
This completes the proof of Lemma 5.

3. Proof of Theorem 1

Suppose that . Since and share and IM, by Lemma 1 we have . Let , , , , and be defined by (11)–(15), respectively. Then from the proof process of Lemma 3 we know , , , , , and . Next we rewrite (13) as where ,  ,  ,  , and are all small functions with respect to .

Now we divide into two cases.

Case 1. ; that is, . We again discuss the three subcases.

Subcase 1. and . Since and share and IM, the zeros of and of multiplicity larger than one are the zeros and , respectively. It then follows that that is,
Let . For , from (13) we get which implies that or .
If , then by (13) we deduce which, in view of the condition that is entire, implies that . From this and Lemma 5, it follows that , contradicting the assumption. Thus . By the conditions in Theorem 1, we know that .
Hence that is, Similarly, It then follows from (44)–(49) and the second fundamental theorem that
For any , from (13) and (14), we can get .
If , then by (13) and (14) we have which implies that where is a nonzero constant. We claim that . Indeed, if , then by (52) we have which leads to . This contradicts the assumption. Thus and so from (52) we get This and Lemma 4 yield which gives . From this and the condition that is entire, it follows that , a contradiction. Hence .
Therefore by (50) and Lemma 3 we obtain which implies that This is impossible by the second fundamental theorem.

Subcase 2. Either and or and . Without loss of generality, we assume that and . It is easy to see by (13) that the zeros of and of multiplicity all larger than one are the zeros of . Thus by Lemma 3, that is, By the discussion of Subcase 1, we see that is, Note that the zeros of of multiplicity larger than one are all the zeros of . Since and share IM, it follows that
Now from (59)–(62) and the second fundamental theorem, we obtain that is, Let It is easily seen from (65) and the lemma of the logarithmic derivative that Note that common simple zeros of and are not the poles of . In terms of the discussion of Subcase 1, from (65) we know , which together with (66) gives that
Let . Then by (65) and (13) we have .
If , then from (65) we derive where is a nonzero constant. This, in view of the condition that is entire, implies that . From this and Lemma 5, it follows that , contradicting the assumption. Thus .
Hence by (64) and (67) we obtain that is, This is also impossible by the second fundamental theorem.

Subcase 3. and . By the discussion of Subcase 2, we see We claim that
Let It is easily known from (74) and the lemma of the logarithmic derivative that Note that common zeros of of multiplicity two and of multiplicity one are not the poles of . In terms of the discussion of Subcase 2, we know , which together with (75) gives that
Let . Then by (74) and (13) we have .
If , then from (76) we get that is,
If , then by (74) we deduce where is a nonzero constant. This implies . Since , , and , we obtain . Thus that is,
Hence (72) follows. In the same manner as above, we can prove (73). The proof of the claim is complete. Now by (71)–(73) we get , a contradiction.

Case 2. . Then by (42) and (i) in Lemma 3 we have which implies that On the other hand, Combining (83) with (84) yields This and Lemma 5 lead to , contradicting the assumption.
This completes the proof of Theorem 1.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referees for their thorough comments and helpful suggestions. Project supported by the National Natural Science Foundation of China (Grant no. 11301076), the Natural Science Foundation of Fujian Province, China (Grant no. 2014J01004), the Education Department Foundation of Fujian Province, China (Grant no. JB13018), and the Innovation Team of Nonlinear Analysis and its Applications of Fujian Normal University, China (Grant no. IRTL1206).