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Abstract and Applied Analysis
Volume 2014, Article ID 615351, 6 pages
http://dx.doi.org/10.1155/2014/615351
Research Article

On the Tumura-Clunie Theorem and Its Application

1School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China
2College of Economics and Management, Zhoukou Normal University, Zhoukou 466001, China

Received 22 January 2014; Revised 15 March 2014; Accepted 26 March 2014; Published 22 April 2014

Academic Editor: Geraldo Botelho

Copyright © 2014 Gaixian Xue and Jinjin Huang. 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 cast aside the restriction of the simple pole in the Tumura-Clunie type theorems for meromorphic functions and obtain a better result which improves the earlier results of Y. D. Ren. Furthermore, as an application, we improve a theorem given by B. Y. Su.

1. Introduction and Main Results

A meromorphic function will always mean meromorphic in the complex plane . We adopt the standard notation in the Nevanlinna value distribution theory of meromorphic functions such as , , , and as explained in [1, 2]. For any nonconstant meromorphic function , we denote by any quantity satisfying as possibly outside a set of finite linear measures that is not necessarily the same at each occurrence.

Definition 1 (see [1]). A meromorphic function “” is said to be a small function of if .

Definition 2. Throughout this paper one denotes by meromorphic functions satisfying . If , we call a polynomial in with degree . If are nonnegative integers, we call a differential monomial in of degree and of weight . If are differential monomials in , we call a differential polynomial in and define the degree and the weight by and , respectively.
Also is called a quasi-differential polynomial generated by if, instead of assuming , we just assume that for the coefficients .

Definition 3. Let be a positive integer; for any in the complex plane, one denotes by the counting function of -points of with multiplicity less than or equal to , by the counting function of -points of with multiplicity more than or equal to , and by the counting function of -points of with multiplicity of . Denote the reduced counting function by , , and , respectively.
Let be a nonconstant meromorphic function and let be a differential polynomial, where is also a differential polynomial and .
Hua (see [3, page 69]) proved the following result.

Theorem A. Let be a nonconstant meromorphic function and let be given by (1) with . If then where is a small function of .
Then , , and is obtained by substituting for , for , and so forth in the terms of degree in .

Remark 4. The conclusion still holds good if condition (2) is replaced with where denotes any quantity which satisfies as through a set of of infinite measure.
Hua (see [3]) improved Theorem A and obtained the following result.

Theorem B. Let be a nonconstant meromorphic function and let be given by (1) with . If then where is a small function of .

Another theorem is due to Zhang and Li (see [4]), which can be stated as follows.

Theorem C. Let be a nonconstant meromorphic function and let be given by (1), where is an integer. Then one of the following occurs.
If , then Or there exists a small proximity function of such that and .
If , then or where is a small function of .
In the special case, if , where , then or where is a small function of .

Corollary 5. From Theorem C we know that if condition (2) is replaced with “” in Theorem A, then the conclusion remains valid.
In this direction Ren (see [5]) also generalized Tumura-Clunie’s theorem concerning differential polynomials.
Combining the methods used in their proofs we show the following theorem.

Theorem 6. Let be a nonconstant meromorphic function and let be given by (1), where is an integer and is the weight of . If then where is a small function of .

It is easily seen from the following example that in Theorem 6 is necessary.

Example 7. Let and . Obviously, (13) is obtained but (14) does not hold.

2. Some Lemmas

To prove our results, we need some lemmas.

Lemma 8 (see [1]). Let and be two nonzero meromorphic functions in the complex plane; then

Lemma 9. If denotes the counting function of those zeros of which are not the zeros of , where a zero of is counted according to its multiplicity, then

Lemma 10. Suppose that is given in Definition 2. Let be a pole of of order and neither a zero nor a pole of coefficients of . Then is a pole of of order at most .

Lemma 11 (see [6]). Let be a nonconstant meromorphic function and let be given in Definition 2. Then

Lemma 12. Suppose that is a nonconstant meromorphic function and is given in Definition 2. Then .

Proof. It is straightforward by Lemma 11.

Lemma 13 (see [7]). Let be a nonconstant meromorphic function in the complex plane and let and be quasi-differential polynomials in . If and , then .

Lemma 14. Let be a nonconstant meromorphic function and let be given by (1). Then

Proof. If , the conclusion of Lemma 14 holds obviously.
In the following we suppose that .
With , we set Let be a simple pole of and not a zero of coefficients of ; then From Lemma 10 we know that is a pole of of order at most ; then we have where .
Then
So . But is a zero of of order at least . Then where denotes the counting function of the zeros of , not of .
By Lemma 8 and Nevanlinna first fundamental theorem, we get From (24), we have From (19), we know that the poles and zeros of can only occur at the multiple zeros of , the zeros of , and the zeros of . Hence where denotes the counting function of the zeros of , not of .
By Lemmas 9 and 12, we obtain Combining (23), (25), (26), and (27), we obtain (18).
This completes the proof of Lemma 14.

Proof of Theorem 6. We consider two cases.
Case 1. If , (14) holds obviously.
Case 2. If , by Lemma 14 and (13) we have This shows that Suppose that .
So we have and ; moreover .
By Lemma 11 we get .
On the other hand, we have It follows that , which is impossible.
Therefore, .
Then
From (29) and the condition of the theorem, we know .
By , we have And hence Let
Then where and are quasi-differential polynomials.
By Lemma 13 we have
By Lemma 10 and (35) we obtain
Note that .
So .
From (34) we know that is a polynomial and .
Set where is a polynomial and is a small function of ; moreover .
Set ; we have where is a polynomial and .
Now proceeding as the above proof, we get
By Lemma 13 we obtain
Therefore we have
Notice that .
We can get .
So , where is a constant. Obviously .
This proves Theorem 6.

3. Application

Very recently, Yi (see [8, 9]) proved the following result.

Theorem D. Let be a transcendental meromorphic function and let be a polynomial, . If and share 0 in , then has infinitely many zeros.

Remark 15. From the hypothesis of Theorem E, it can be easily seen that all zeros of have multiplicity at least two.
Ren and Yang 2013 (see [10]) obtained the following result.

Theorem E. Let be a transcendental meromorphic function and let be a rational function, . Suppose that, with the exception of possibly finitely many, all zeros and poles of are multiple. Then has infinitely many zeros.

It is natural to ask the following question: what can we say if is replaced by and and are replaced by a small function relative to in Theorems D and E?

Later, Yang (see [11]) answered the above question and obtained the following result.

Theorem F. Let be a transcendental meromorphic function satisfying Then, for any and any small function of ,

We supplement Theorems D and E, improve Theorem F, and obtain the following result.

Theorem 16. Let be a transcendental meromorphic function satisfying Then, for any and any small function of ,

The method of our proof essentially belongs to Yang. For the completeness, we give the proof here.

Proof. Set
Then
Obviously
Now
Thus, in general, where denotes a homogeneous differential polynomial in of degree . So
If the assertion of the theorem was false, that is,
then from (52) we have
Thus from (48), (53), and (54), we obtain
Combining Theorem 6, (55) gives where (a small function of ) is determined by the two equations: and .
We may claim that(i);(ii);(iii) for all .
In fact, from the definition of we know that the claim (i) above holds.
By (54) we have .
From , , and (29) we get
That is, the claim (ii) above holds.
Combining (53) and the claims (i) and (ii), we may deduce
Then the claim (iii) is true also.
Thus, by (54) and (56), we obtain
Since , it follows that which is impossible unless .
But then, from (59), and we have which contradicts the fact that is a transcendental meromorphic function.
This completes the proof of Theorem 16.

Remark 17. For , from the proof of Theorem 16 and Corollary 5, we know that if the condition “” is replaced with “” in Theorem 16, then the conclusion still holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11301140 and U1304102.

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