Abstract and Applied Analysis

Volume 2014 (2014), Article ID 894968, 7 pages

http://dx.doi.org/10.1155/2014/894968

Research Article

Unicity of Meromorphic Functions Sharing Sets with Their Linear Difference Polynomials

Sheng Li and BaoQin Chen

College of Science, Guangdong Ocean University, Zhanjiang 524088, China

Received 23 January 2014; Accepted 20 March 2014; Published 13 April 2014

Academic Editor: Zhi-Bo Huang

Copyright © 2014 Sheng Li and BaoQin Chen. 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 mainly investigate the unicity of meromorphic functions sharing two or three sets with their linear difference polynomials and prove some results.

1. Introduction and Main Results

In this paper, we assume the reader is familiar with the fundamental results and the basic notations of the Nevanlinna theory of meromorphic functions (see, e.g., [1–3]). Let be meromorphic in the whole plane. We use the notation to denote the order of growth of the meromorphic function . In addition, we denote by any quantity satisfying , as outside of a possible exceptional set of finite logarithmic measure. We say that a meromorphic function is a small function of provided that . Let be the set of all small functions of .

For a set , we define the following:

Let and be meromorphic functions. If and , respectively, then we say that and share a set CM and IM, respectively.

Furthermore, let be a nonzero complex constant. We define the shift of by , and define the difference operators of by

The unicity theory of meromorphic functions sharing sets is an important topic of the uniqueness theory. First of all, we recall the following theorem given by Li and Yang in [4].

Theorem A (see [4]). Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Then, for any two nonconstant meromorphic functions and , the conditions and imply .

Yi and Lin considered the case with the condition that two meromorphic functions share three sets and got the result as follows.

Theorem B (see [5]). Let , , and , where , are nonzero constants such that has no repeated root and is an integer. If, for two nonconstant meromorphic functions and , for , and , then .

Recently, a number of papers have focused on difference analogues of the Nevanlinna theory (see, e.g., [6–9]). In particular, there has been an increasing interest in studying the uniqueness problems related to meromorphic functions and their shifts or their difference operators (see, e.g., [10–16]).

In 2010, Zhang considered a meromorphic function sharing sets with its shift and proved the following result.

Theorem C (see [16]). Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Suppose that is a nonconstant meromorphic function of finite order. Then and imply .

For an analogue result in difference operator, B. Chen and Z. Chen proved the following theorem in [10].

Theorem D (see [10]). Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Suppose that is a nonconstant meromorphic function of finite order satisfying and . If

then .

It is natural to ask what happens if the shift or difference operator is replaced by a general expression of , such as a linear difference polynomial of .

Here, a linear difference polynomial of is an expression of the form

where , are small functions of , are complex constants, and is a nonnegative integer.

In this paper, our aim is to investigate the uniqueness problems of linear difference polynomials of . In particular, we primarily consider the linear difference polynomial which satisfies one of the following conditions:

Corresponding to the above question, we obtain the following results.

Theorem 1. Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Suppose that is a nonconstant meromorphic function of finite order and is of the form (4) satisfying the condition in (5). If and , then .

Corollary 2. Let , , and be given as in Theorem 1. Suppose that is a nonconstant meromorphic function of finite order satisfying the following:

If and , then .

With an additional restriction on the order of growth of , we prove the following fact.

Theorem 3. Let , , and be given as in Theorem 1. Suppose that is a nonconstant meromorphic function of finite order such that . If and , then .

Remark 4. Note that, in Theorem 3, we do not assume that the linear polynomial satisfies the condition in (5). In fact, since , by (19), we can easily get , which implies . Then using a similar method as in the proof of Theorem 1, we can complete the proof of Theorem 3.

Now we may ask what happens if the condition in Theorem 1 is replaced by a weaker condition containing the case or even . By considering three sets, we get the following theorem.

Theorem 5. Let , be nonnegative integers such that . Let and be nonzero constants such that has no multiple roots. Let , , and . Suppose that is a nonconstant meromorphic function of finite order, is of the form (4) satisfying the condition in (5), and for . Then one has the following.

(i)If , then , where .(ii)If and are coprime, then .

Remark 6. Taking in Theorem 5, we can obtain an analogue result of Theorem B related to linear difference polynomials.

Furthermore, the following result is a corollary of Theorem 5 related to difference operators.

Corollary 7. Let , , and , , be given as in Theorem 5. Suppose that is a nonconstant meromorphic function of finite order satisfying

and for . Then one has the following.

(i)If , then , where .(ii)If and are coprime, then .

Finally, we give some examples for our results.

Examples. In the following, let be an entire function with period 1 such that (see [17]).

(1)For the case (i) of condition (5), let , , and let . Then for , and the sum of the coefficients of is equal to . These examples satisfy Theorems 1 and 5 but do not satisfy Theorem D.(2)For the case (ii) of condition (5), let and let . Then , the sum of the coefficients of equals , and

 This example satisfies Theorems 1 and 5 and Corollaries 2 and 7.(3)For Theorem 3, let and let . Then and the sum of the coefficients of equals . This example satisfies Theorem 3 but does not satisfy Theorem D and Theorems 1 and 5.

2. Proof of Theorem 1

We need the following lemmas for the proof of Theorem 1.

The difference analogue of the logarithmic derivative lemma was given by Halburd-Korhonen [7] and Chiang-Feng [6] independently. We recall the following lemmas.

Lemma 8 (see [7]). Let be a nonconstant meromorphic function of finite order, and . Then

for all outside of a possible exceptional set with finite logarithmic measure.

Lemma 9 (see [8]). Let , let , and let be a meromorphic function of finite order. Then for any small periodic function with period , consider the following:

where the exceptional set associated with is of at most finite logarithmic measure.

Let be a meromorphic function of finite order. Notice that if (≢0) is of the form (4) such that , then, for any given complex constant , . This indicates that and hence

With this, one can easily prove Lemma 10 below by a similar reasoning as in the proof of the difference analogue of the second main theorem of the Nevanlinna theory in [8] by Halburd and Korhonen. We omit those details.

Lemma 10. Let , let be a meromorphic function of finite order, and let be of the form (4) such that . Let and let be distinct complex constants. Then

where

and the exceptional set associated with is of at most finite logarithmic measure.

Remark 11. If the linear difference polynomial is replaced by

Lemma 10 also holds even if the distinct complex constants are replaced by which are distinct meromorphic periodic functions with period such that for all .

The following is the standard Valiron-Mohon’ko theorem; (see Theorem 2.2.5 in the book of Laine [2]).

Lemma 12 (see [2]). Let be a meromorphic function. Then, for all irreducible rational functions in ,

with meromorphic coefficients , such that

The characteristic function of satisfies

where .

Proof of Theorem 1. Since and share CM, we see that and . Then by Lemma 8, we have

Since , where and the equation has no multiple roots, we know that and share 0 CM. Then from this and the condition , there exists a polynomial such that

Suppose that . Note that and the equation has no multiple roots. Let denote all different roots of the equation .

Next we prove that . We know that

(i)If , we see that

Then we deduce from this, (19), and Lemma 9 that

(ii)If , we have

From this, (19), and Lemma 9, we get

Applying Lemma 10 to , we get

Then the assumptions in (5), (24), and (25) yield the following:

To sum up, we now prove that . Rewriting (19), we get

Denote . It follows from Lemma 12 and that

Hence, .

By (18) and (27) and applying the second main theorem for three small target functions, we deduce the following:

By combining (28) and (29), we have

which contradicts with .

Now we turn to consider the case . Equation (19) yields the following:

Set , and we have

If is not a constant, (32) can be rewritten as

where and .

By the assumption that and have no common factors, we see that , are different. Assume that is a -point of of multiplicity , where . Notice that

is a constant. Then (33) implies that is a pole of . Thus, . This yields the following, for :

Then by (35), we get

which is impossible with and .

Hence, is a constant. Since is a nonconstant meromorphic function, we deduce from (32) that . This yields , which completes the proof of Theorem 1.

3. Proof of Theorem 5

Since is a nonconstant meromorphic function of finite order, for , , , and , we have , , and , and we also get (18) and (19).

Since and share , CM, there exists a polynomial such that

By Lemma 8, we see that

As in the proof of Theorem 1, we see that still holds in both cases (i) and (ii).

Rewriting (19), we have

Combining this and (37), we get

Suppose that . If , (40) becomes

By the condition that , implies .

It follows from (38), (41), and that

which is a contradiction, since .

If , it follows from (38), (41), and that

That is impossible.

Therefore, . Notice that . Using a similar method, we can prove that . Then (40) implies that .

If , we have . Obviously, is a constant. Set . Thus, by (37), we get , where .

If and are coprime, and imply that . Thus, by (37), we get . Thus, Theorem 5 is proved.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Both authors drafted the paper and read and approved the final paper.

Acknowledgments

The authors are grateful to the editor and referees for their valuable suggestions. This work was supported by the NNSFC (no. 11171119, 11301091), the Guangdong Natural Science Foundation (no. S2013040014347), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. 2013LYM_0037).

References

W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.

I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993.

C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

P. Li and C.-C. Yang, “Some further results on the unique range sets of meromorphic functions,” Kodai Mathematical Journal, vol. 18, no. 3, pp. 437–450, 1995.

H.-X. Yi and W.-C. Lin, “Uniqueness theorems concerning a question of Gross,” Proceedings of the Japan Academy, Series A. Mathematical Sciences, vol. 80, no. 7, pp. 136–140, 2004.

Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008.

R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006.

R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 31, no. 2, pp. 463–478, 2006.

I. Laine and C.-C. Yang, “Clunie theorems for difference and -difference polynomials,” Journal of the London Mathematical Society, vol. 76, no. 3, pp. 556–566, 2007.

B. Chen and Z. Chen, “Meromorphic function sharing two sets with its difference operator,” Bulletin of the Malaysian Mathematical Sciences Society. Second Series, vol. 35, no. 3, pp. 765–774, 2012.

B. Chen, Z. Chen, and S. Li, “Uniqueness theorems on entire functions and their difference operators or shifts,” Abstract and Applied Analysis, vol. 2012, Article ID 906893, 8 pages, 2012.

Z.-X. Chen and H.-X. Yi, “On sharing values of meromorphic functions and their differences,” Results in Mathematics, vol. 63, no. 1-2, pp. 557–565, 2013.

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, “Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 352–363, 2009.

S. Li and B. Chen, “Meromorphic functions sharing small functions with their linear difference polynomials,” Advances in Difference Equations, vol. 2013, article 58, 2013.

S. Li and Z. Gao, “Entire functions sharing one or two finite values CM with their shifts or difference operators,” Archiv der Mathematik, vol. 97, no. 5, pp. 475–483, 2011.

J. Zhang, “Value distribution and shared sets of differences of meromorphic functions,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 401–408, 2010.

M. Ozawa, “On the existence of prime periodic entire functions,” Kōdai Mathematical Seminar Reports, vol. 29, no. 3, pp. 308–321, 1978.