Uniqueness of an entire function sharing a small function with its linear differential polynomial with non-constant coefficients

The uniqueness problems of entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In our paper, we study the uniqueness of an entire function when it shares a small function with its first derivative and two linear differential polynomials of different orders. Here we consider the differential polynomial with non-constant coefficients. In particular, the result of the paper improves the results due to P. Li [7], I. Kaish and Md. M. Rahaman [4].