ON A GENERALIZED BRIOT–BOUQUET TYPE DIFFERENTIAL SUBORDINATION

Abstract

We introduce and study the following special type of differential subordination implication:

$$p(z)Q(z)+\frac{z{p}^{\prime }(z)}{\beta p(z)+\alpha }\prec h(z)\Rightarrow p(z)\prec h(z),$$

which generalizes the Briot–Bouquet differential subordination, where $Q(z)$ is analytic and $0\ne \beta ,\alpha \in ℂ$. Further, we discuss some special cases of our result. Finally, we obtain analogues of the open door lemma and the integral existence theorem with applications to univalent functions.