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Unicity of Meromorphic Function Sharing One Small Function with Its Derivative
We deal with the problem of uniqueness of a meromorphic function sharing one small function with its k's derivative and obtain some results.
1. Introduction and Main Results
In this article, a meromorphic function means meromorphic in the open complex plane. We assume that the reader is familiar with the Nevanlinna theory of meromorphic functions and the standard notations such as and so on.
Let and be two nonconstant meromorphic functions; a meromorphic function is called a small functions with respect to provided that Note that the set of all small function of is a field. Let be a small function with respect to and We say that and share CM(IM) provided that and have same zeros counting multiplicities (ignoring multiplicities).
Moreover, we use the following notations.
Let be a positive integer. We denote by the counting function for the zeros of with multiplicity and by the corresponding one for which the multiplicity is not counted. Let be the counting function for the zeros of with multiplicity , and let be the corresponding one for which the multiplicity is not counted. Set And we define
Brück (see ) considered the uniqueness problems of an entire function sharing one value with its derivative and proved the following result.
Theorem 1 A. Let be nonconstant entire function. If and share the value CM and if , then for some constant .
Theorem 1 B (see). Let be a nonconstant meromorphic function and, let be a positive integer. Suppose that and share CM and for where is a set of infinite linear measure and satisfies then for some constant
Theorem 1 C (see). Let be a nonconstant, nonentire meromorphic function and be a small function with respect to If (1) and have no common poles,(2) and share the value CM,(3) then where is a positive integer.
In the same paper, Yu  posed four open questions. Lahiri and Sarkar  and Zhang  studied the problem of a meromorphic or an entire function sharing one small function with its derivative with weighted shared method and obtained the following result, which answered the open questions posed by Yu .
Theorem 1 D (see). Let be a non-constant meromorphic function and, let be a positive integer. Also let be a meromorphic function such that Suppose that and share IM and for and is a set of infinite linear measure. Then for some constant .
In this article, we will pay our attention to the value sharing of and that share a small function and obtain the following results, which are the improvements and complements of the above theorems.
Theorem 1.1. Let 11 be integers and let be a non-constant meromorphic function. Also let be a small function with respect to . If and share IM and or and share CM and for , , and is a set of infinite linear measure, then for some constant
Theorem 1.2. Let 11 be integers and be a non-constant meromorphic function. Also let be a small function with respect to . If and share IM and or and share CM and then .
2. Some Lemmas
In this section, first of all, we give some definitions which will be used in the whole paper.
Definition 2.1. Let and be two meromorphic functions defined in assume, that and share IM; let be a zero of with multiplicity and a zero of with multiplicity We denote bythe counting function of the zeros of where and bythe counting function of zeros of where We denotes by the counting function of the zeros of where each point is counted according to its multiplicity, and denote its reduced form. In the same way, we can define, and so on.
Definition 2.2. In this paper denotes the counting function of the zeros of which are not the zeros of and and denotes its reduced form. In the same way, we can define and
Next we present some lemmas which will be needed in the sequel. Let be two non-constant meromorphic functions defined in We shall denote by the following function:
Lemma 2.3 (see). Let be two nonconstant meromorphic functions defined in If and are sharing IM, then If and are sharing CM, then
Lemma 2.4 (see). Let be a meromorphic function and is a finite complex number. Then (i)(ii) for (iii)where are two meromorphic functions such that
Lemma 2.5 (see). Let be a non-constant meromorphic function, and are two positive integers. Then
Lemma 2.6 (see). Let be a non-constant meromorphic function and let be a positive integer. where are meromorphic functions such that and Then
3. Proof of Theorem 1.1
From the definitions of and recalling that and share value IM(CM), we get
We will distinguish two cases below.
Case 1 (). From (2.1) it is easy to see that Subcase 1.1. Suppose that and share IM. According to (3.1), and share IM except the zeros and poles of By (3.1), we have
Let be a simple zero of and but Through a simple calculation we know that is a zero of so
From (3.4)–(3.6) and Lemma 2.3, we have It follows by the second fundamental theorem, (3.5), and (3.7) that By Lemma 2.5, we have which contradicts (1.4).Subcase 1.2. Suppose that and share CM.
Let be a simple zero of and , but By a simple calculation, we can still get Therefore
Noting that by (3.4) and Lemma 2.3, we can deduce
By the second fundamental theorem, (3.5), and (3.11), we have
Taking into account (3.1), we have
This contradicts (1.5).
Case 2 (). Integration yields
where are constants and It is easy to see that and share CM. Now we claim that
If then by (3.14) we get So our claim holds. Hence we can assume that If then we can rewrite (3.14) as So If then by Lemma 2.4 and (3.17) we have Hence that is, This is a contradiction with (1.4) and (1.5). If then from (3.14) we get We rewrite it as So by Lemmas 2.4 and 2.6 and (3.15), we have This implies that , since . This is impossible. Hence our claim is right. So . Theorem 1.1 is, thus, completely proved.
4. Proof of Theorem 1.2
Case 1. Subcase 1.1. Suppose that and share IM, then we can still get (3.6) and (3.7). Then by the second fundamental theorem, Lemma 2.3, and (3.5) we have Applying Lemma 2.5 to the above inequality and noticing the definition of we get This implies that This contradicts (1.6).Subcase 1.2. Suppose that and share CM. Similarly as above, we can easily obtain by Lemma 2.3, we can deduce So by the second fundamental theorem, (4.4), and using Lemma 2.5 again, we have This implies that This contradicts (1.7).
Case 2 (). Similarly, we can also get (3.14). Next we claim that . If then it follows that from (3.14). Hence, we may assume that (3.15) holds. If then
Again by second fundamental theorem and (4.4) we have that is, Then we have and it follows that and from (3.15) we have then with (1.6) and (1.7) we may deduce It is impossible, and we can assume that thus, we can get It shows that
If , by (4.11), then we have which with the above equality may lead to which is impossible. If then by second fundamental theorem, Lemma 2.5, (3.15), and (4.11) we have which with (3.15) may deduce so which with and (1.6) may deduce which is impossible. Hence our claim holds.
Next we will prove that From (3.17) we have Then If then we have By Lemma 2.5, we get It implies that Combining (4.16) with (1.6) yields that is, This is a contradiction.
Combining (4.16) with (1.7) yields that is, , which is also a contradiction. Hence and Now Theorem 1.2 has been completely proved.
The authors would like to express their sincere thanks to the referee for helpful comments and suggestions.
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