Non-linear difference polynomials sharing a polynomial with finite weight

Abstract

The uniqueness theory of meromorphic function mainly studies the conditions under which there exists only one function satisfying these conditions. The uniqueness theory of entire and meromorphic functions has grown up as an extensive sub-field of value distribution theory and the Nevanlinna's Five value and Four value theorems serves as the starting point of this uniqueness theory. In this paper, we consider a linear difference polynomial $\mathcal{L}_{\eta}(\mathfrak{f})=\mathfrak{f}(z+\eta)+\eta_0\mathfrak{f}(z)$, of the finite ordered non-constant meromorphic function $\mathfrak{f}$, with $\eta$ and $\eta_0$ being finite non-zero complex constants, and with the help of Nevanlinna theory, we analyse the uniqueness results between two finite ordered non-constant meromorphic functions $\mathfrak{f}$ and $\mathfrak{g}$, when their non-linear difference polynomials $\mathfrak{f}^n(z)\mathcal{L}_{\eta}(\mathfrak{f})$ and $\mathfrak{g}^n(z)\mathcal{L}_{\eta}(\mathfrak{g})$, with $n \ge 2$ being a positive integer shares a non-zero polynomial $p(z)$ with finite weights 0,1 and 2. Our results extend and improve some of the earlier results of Majumder (\textit{Applied Mathematics E-Notes, (17): 114-123, 2017)