  We introduce and study the following special type of differential
subordination implication: \begin{equation}\label{abs}
p(z)Q(z)+\frac{zp'(z)}{\beta p(z)+\alpha}\prec h(z)\quad\Rightarrow p(z)\prec
h(z), \end{equation} which involves generalization of the Briot-Bouquet
differential subordination, where $Q(z)$ is analytic and
$0\neq\beta,\alpha\in\mathbb{C}.$ Further, some special cases of our result are
also discussed. Finally, analogues of open door lemma and integral existence
theorem with applications to univalent functions are obtained.
