- Open Access
Uniqueness problems on entire functions that share a small function with their difference operators
© Chen and Li; licensee Springer. 2014
- Received: 26 July 2014
- Accepted: 25 November 2014
- Published: 8 December 2014
In this paper, we consider uniqueness problems on entire functions that share a small periodic entire function with their two difference operators and obtain some results. Our first theorem provides a difference analogue of a result of Li and Yang (J. Math. Anal. Appl. 253(1):50-57, 2001).
- entire functions
- difference operators
Throughout this paper, a meromorphic function always means meromorphic in the whole complex plane, and c always means a nonzero constant. We use the basic notations of the Nevanlinna theory of meromorphic functions such as , , and as explained in [1–3]. In addition, we say that a meromorphic function is a small function of if , where , as outside of a possible exceptional set of finite logarithmic measure.
In particular, for the case .
Let and be two meromorphic functions, and let be a small function of and . We say that and share IM, provided that and have the same zeros ignoring multiplicities. Similarly, we say that and share CM, provided that and have the same zeros counting multiplicities.
The problem on meromorphic functions sharing small functions with their derivatives is an important topic of uniqueness of meromorphic functions.
In 1986, Jank, Mues and Volkmann  proved the following result.
Theorem A ()
Let f be a nonconstant meromorphic function, and let be a finite constant. If f, , and share the value a CM, then .
Many authors have been considering about some related cases, and got some interesting results (see, e.g., [5, 6]). In 2001, Li and Yang  obtained the following result for a special case that is an entire function, and f, , and share one value.
Theorem B ()
where b, c are nonzero constants and .
Recently, a number of papers (including [7–12]) have focused on difference analogues of Nevanlinna theory. In addition, many papers have been devoted to the investigation of the uniqueness problems related to meromorphic functions and their shifts or their difference operators and got a lot of results (see, e.g., [13–15]).
Our aim in this paper is to investigate uniqueness problems on entire functions that share a small periodic entire function with their two difference operators and provide a difference analogue of Theorem B. We now state the following theorem, which is the main result of this paper.
Theorem 1.1 Let be a nonconstant entire function of finite order, and let be a periodic entire function with period c. If , , and () share CM, then .
Let , then , and hence , Δf, and share 1 CM, but . This example shows that the conclusion in Theorem 1.1 cannot be extended to in general.
Let , then , , and hence , Δf and share 0 CM, but (). This example shows that the restriction in Theorem 1.1 is necessary.
Remark In the above example (1), can be changed to , where is a periodic entire function with period 1, and the result still holds. This shows that the order of the function in Theorem 1.1 is not always one.
As a continuation of Theorem 1.1 and example (2) above, we prove the following result.
Theorem 1.2 Let be a nonconstant entire function of finite order. If , , and () share 0 CM, then , where C is a nonzero constant.
Firstly, we present some lemmas which will be needed in the proof of Theorem 1.1.
Lemma 2.1 ()
where the exceptional set associated with is of at most finite logarithmic measure.
Lemma 2.2 ()
the orders of are less than that of for , ,
where and are polynomials.
Thus, by (2.3) and (2.4), we have . Similarly, .
Now we divide this proof into the following two steps.
where , , .
Moreover, is a polynomial of and its shifts .
Here , for , , are polynomials with degree less than m.
That is impossible.
Now we distinguish three cases as follows:
By Lemma 2.2, we have , which is impossible.
Case (ii). Suppose that . Then, by a similar argument to above, we can also get a contradiction.
By this together with (2.14), (2.15), (2.16), and Lemma 2.2, we can get a contradiction.
By a similar method as the above, we can also get . That is impossible.
where are polynomials with degree less than .
By this, together with (2.20) and Lemma 2.2, we obtain , which is impossible.
We get a contradiction again.
Hence, . By (2.24), we see that , which implies . That is impossible.
which implies . That is impossible.
Hence, we must have , and Theorem 1.1 is proved. □
where and are polynomials.
This completes our proof.
If is not a constant, by assuming that is not a constant, with a similar arguing as in the proof of Theorem 1.1, we can deduce that the case is impossible.
This indicates that is a zero of of order at least , which is impossible. Theorem 1.2 is thus proved.
This work was supported by the NNSFC (No. 11301091) and the Guangdong Natural Science Foundation (No. S2013040014347) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (No. 2013LYM_0037).
- Hayman WK Oxford Mathematical Monographs. In Meromorphic Functions. Clarendon, Oxford; 1964.Google Scholar
- Laine I de Gruyter Studies in Mathematics 15. In Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar
- Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.View ArticleMATHGoogle Scholar
- Jank G, Mues E, Volkmann L: Meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen. Complex Var. Theory Appl. 1986, 6(1):51-71. 10.1080/17476938608814158MathSciNetView ArticleMATHGoogle Scholar
- Li P, Yang CC: Uniqueness theorems on entire functions and their derivatives. J. Math. Anal. Appl. 2001, 253(1):50-57. 10.1006/jmaa.2000.7007MathSciNetView ArticleMATHGoogle Scholar
- Yang LZ: Further results on entire functions that share one value with their derivatives. J. Math. Anal. Appl. 1997, 212: 529-536. 10.1006/jmaa.1997.5528MathSciNetView ArticleMATHGoogle Scholar
- Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133-147. 10.1017/S0305004106009777MathSciNetView ArticleMATHGoogle Scholar
- Chen ZX: Relationships between entire functions and their forward difference. Complex Var. Elliptic Equ. 2013, 58(3):299-307. 10.1080/17476933.2011.584251MathSciNetView ArticleMATHGoogle Scholar
- Chen ZX, Shon KH: Properties of differences of meromorphic functions. Czechoslov. Math. J. 2011, 61: 213-224. 10.1007/s10587-011-0008-zMathSciNetView ArticleMATHGoogle Scholar
- Chiang YM, Feng SJ:On the Nevanlinna characteristic of and difference equations in the complex plane. Ramanujan J. 2008, 16(1):105-129. 10.1007/s11139-007-9101-1MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463-478.MathSciNetMATHGoogle Scholar
- Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314(2):477-487. 10.1016/j.jmaa.2005.04.010MathSciNetView ArticleMATHGoogle Scholar
- Chen BQ, Chen ZX, Li S: Uniqueness theorems on entire functions and their difference operators or shifts. Abstr. Appl. Anal. 2012., 2012: Article ID 906893Google Scholar
- Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang J: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 2009, 355: 352-363. 10.1016/j.jmaa.2009.01.053MathSciNetView ArticleMATHGoogle Scholar
- Zhang JL: Value distribution and shared sets of differences of meromorphic functions. J. Math. Anal. Appl. 2010, 367: 401-408. 10.1016/j.jmaa.2010.01.038MathSciNetView ArticleMATHGoogle Scholar
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