Value distribution of meromorphic functions concerning differences

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 \(1 \le \rho (f) < \infty \) $1\le \rho (f)<\infty$ , let c be a nonzero constant, n a positive integer, and let P, Q be two polynomials. If \(\max \left\{ \lambda (f-P), \lambda \left( \frac{1}{f}\right) \right\} <\rho (f)\) $max\left\{\lambda ,(f-P),,,\lambda ,\left(\frac{1}{f}\right)\right\}<\rho (f)$ and \(\Delta _{c}^{n}f \not \equiv 0\) ${\mathrm{\Delta }}_c^{n}f\not\equiv 0$ , then we have (i) \(\delta (Q, \Delta _c^n f)=0\) $\delta (Q,{\mathrm{\Delta }}_c^{n}f)=0$ and \(\lambda (\Delta _{c}^{n}f-Q)=\rho (f)\) $\lambda ({\mathrm{\Delta }}_c^{n}f-Q)=\rho (f)$ , for \(\Delta _{c}^{n}P\not \equiv Q\) ${\mathrm{\Delta }}_c^{n}P\not\equiv Q$ ; (ii) \(\delta (Q, \Delta _c^n f)=1\) $\delta (Q,{\mathrm{\Delta }}_c^{n}f)=1$ and \(\lambda (\Delta _{c}^{n}f-Q)<\rho (f)\) $\lambda ({\mathrm{\Delta }}_c^{n}f-Q)<\rho (f)$ , for \(\Delta _{c}^{n}P\equiv Q\) ${\mathrm{\Delta }}_c^{n}P\equiv Q$ . 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].