Entire function with its linear differential polynomial normally or weakly shares a set and some related topics

We strive to find different forms of an entire function sharing a doubleton of small functions CM with its linear differential polynomial to extend a result of Li [J Math Anal Appl 310:412–423, 2005]. In our another attempt, we have conveniently first time answered an open question raised in Li (2005) under weakly sharing environment. We have also improved a very recent result of Lahiri [Comput Methods Funct Theo 21:379–397, 2021] by responding an open question of the same. Finally, in the concluding section we have exhibited a two fold improvement of a recent result of Qi et al. [Comput Methods Funct Theo 18:567–582, 2018] which can somehow been interlinked to one of our theorem in the previous section. We also put two open questions and a handful number of examples relevant to the content of the paper.