On the study of Brück conjecture and some non-linear complex differential equations

In this paper, we prove the following result: Let f(z) and α(z) be two non-constant entire functions satisfying σ(α)<μ(f) and P(z) be a polynomial. If f is a non-constant entire solution of the differential equation M[f]+β(z)−α(z)=(fγM−α(z))eP(z), where β(z) is an entire function satisfying σ(β)<μ(f). Then σ2(f)=degP. Our result generalizes the results due to Gundersen and Yang, Chang and Zhu and Li and Cao.