On a Generalized Briot-Bouquet type Differential Subordination

TL;DR

This paper introduces a generalized form of Briot-Bouquet differential subordination involving complex parameters, providing new implications, special cases, and applications to univalent functions.

Contribution

It generalizes existing differential subordination results by incorporating complex parameters and derives new implications, lemmas, and theorems with applications to univalent functions.

Findings

Established a new differential subordination implication involving complex parameters.

Derived analogues of open door lemma and integral existence theorem.

Applied results to the theory of univalent functions.

Abstract

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.