Uniqueness results for a nonlinear differential polynomial (Q432533)

The uniqueness of two meromorphic functions when some nonlinear differential polynomials generated by them share a nonzero value or a small function is a topic of current interest among researchers in function theory. Different kinds of nonlinear differential polynomials are considered by various authors. \textit{A. Banerjee} and \textit{S. Mukherjee} [Arch. Math., Brno 44, No. 1, 41--56 (2008); corrigendum ibid. 44, No. 4, 335--337 (2008; Zbl 1212.30113)] considered weighted sharing of zeros of \(f^{n}(af^{2} + bf + c)f' - \alpha\) and \(g^{n}(ag^{2} + bg + c)g' - \alpha\), where \(f\), \(g\) are meromorphic functions, \(a, b, c\) are constants and \(\alpha \not\equiv 0, \infty\) is a small function of \(f\) ang \(g\). In the paper under review the authors improve the result of Banerjee and Mukherjee by considering the zero sharing of \(f^{n}(f^{k_{1}} + af^{k_{2}} + b)f' - \alpha\) and \(g^{n}(g^{k_{1}} + ag^{k_{2}} + b)g' - \alpha\), where \(n > k_{1} + 2\) and \(k_{1} > k_{2}\) are three positive integers and \(a, b\) are constants.