Normality and uniqueness of homogeneous differential polynomials

The primary goal of this work is to determine whether the results from [19, 20] still hold true when a differential polynomial is considered in place of a differential monomial. In this perspective, we continue our study to establish the uniqueness theorem for homogeneous differential polynomial of an entire and its higher order derivative sharing two polynomials using normal family theory as well as to obtain normality criteria for a family of analytic functions in a domain concerning homogeneous differential polynomial of a transcendental meromorphic function satisfying certain conditions. Meanwhile, as a result of this investigation, we proved three theorems that provide affirmative responses for the purpose of this study. Several examples are offered to demonstrate that the conditions of the theorem are necessary.