Meromorphic Function Sharing Sets with Its Difference Operator or Shifts

Abstract

Let f be a nonconstant meromorphic function, c ∊ ℂ, and let a(z)(≢ 0) ∊ S(f) be a meromorphic function. If f(z) and P(z, f(z)) share the sets {a(z), −a(z)}, {0} CM almost and share {∞} IM almost, where P(z, f(z)) is defined as (1.1), then f(z) ≡ ±P(z, f(z)) or f(z)P(z, f(z)) ≡ ±a ²(z). This extends the results due to Chen and Chen (2013), Liu (2009) and Yi (1987).

Keywords

Meromorphic function / Difference operator / Shared sets

References

[1] Chen B Q, Chen Z X. Entire functions sharing sets of small functions with their difference operators or shifts. Math. Sloυaca, 2013, 6: 1233-1246

[2] Chiang Y M, Feng S J. On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane. Ramanujan J., 2008, 16: 105-129

[3] Fang M L. Unicity theorem or meromorphic function and its differential polynomial. Adv. in Math. (PRC), 1995, 24: 244-249

[4] Gross F, Osgood C F. Entire Functions with Common Preimages, Factorization Theory of Meromorphic Functions and Related Topics. Lect. Notes Pure Appl. Math., 1982, New York: Marcel Dekker 19-24 78

[5] Halburd R G, Korhonen R J. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl., 2006, 314: 477-487

[6] Hayman W K. Meromorphic Functions, 1964, Oxford: Clarendon Press

[7] Liu K. Meromorphic functions sharing a set with applications to difference equations. J. Math. Anal. Appl., 2009, 358: 384-393

[8] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions, 2003, Beijing: Science Press

[9] Yi H X. Meromorphic functions with common preimages. J. of Math. (PRC), 1987, 3: 219-224

[10] Zhang J. Value distribution and shared sets of differences of meromorphic functions. J. Math. Anal. Appl., 2010, 367: 401-408