  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 generalizes 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.
