Value distribution of meromorphic functions concerning differences

Abstract

In this paper, we study value distribution of meromorphic functions concerning differences and mainly prove the following result: Let f be a transcendental meromorphic function of \(1 \le \rho (f) < \infty \), let c be a nonzero constant, n a positive integer, and let P, Q be two polynomials. If \(\max \left\{ \lambda (f-P), \lambda \left( \frac{1}{f}\right) \right\} <\rho (f)\) and \(\Delta _{c}^{n}f \not \equiv 0\), then we have (i) \(\delta (Q, \Delta _c^n f)=0\) and \(\lambda (\Delta _{c}^{n}f-Q)=\rho (f)\), for \(\Delta _{c}^{n}P\not \equiv Q\); (ii) \(\delta (Q, \Delta _c^n f)=1\) and \(\lambda (\Delta _{c}^{n}f-Q)<\rho (f)\), for \(\Delta _{c}^{n}P\equiv Q\). The results obtained in this paper extend and improve some results due to Chen-Shon[J Math Anal Appl 2008], [Sci China Ser A 2009], Liu[Rocky Mountain J Math 2011], Cui-Yang[Acta Math Sci Ser B 2013], Chen[Complex Var Elliptic Equ 2013], Wang-Liu-Fang[Acta Math. Sinica (Chinese Ser) 2016].