Briot-Bouquet differential subordination and Bernardi's integral operator

TL;DR

This paper establishes parameter conditions for analytic functions to be subordinate to specific functions under differential subordination, and explores Bernardi's integral operator's role in generating subclasses of starlike functions.

Contribution

It provides new parameter conditions for differential subordination involving Bernardi's integral operator and specific target functions.

Findings

Derived conditions for subordination to +Az)/(1+Bz)

Established criteria for subordination to +z and e^{z}

Obtained sufficient conditions for Bernardi's operator to produce starlike functions

Abstract

The conditions on $A$, $B$, $\beta$ and $\gamma$ are obtained for an analytic function $p$ defined on the open unit disc $\mathbb{D}$ and normalized by $p(0)=1$ to be subordinate to $(1+Az)/(1+Bz)$, $-1\leq B<A \leq 1$ when $p(z)+ zp'(z)/(\beta p(z)+\gamma)$ is subordinate to $e^{z}$. The conditions on these parameters are derived for the function $p$ to be subordinate to $\sqrt{1+z}$ or $e^{z}$ when $p(z)+ zp'(z)/(\beta p(z)+\gamma)$ is subordinate to $(1+Az)/(1+Bz)$. The conditions on $\beta$ and $\gamma$ are determined for the function $p$ to be subordinate to $e^{z}$ when $p(z)+ zp'(z)/(\beta p(z)+\gamma)$ is subordinate to $\sqrt{1+z}$. Related result for the function $p(z)+ zp'(z)/(\beta p(z)+\gamma)$ to be in the parabolic region bounded by the $\operatorname{Re} w=|w-1|$ is investigated. Sufficient conditions for the Bernardi's integral operator to belong to the various subclasses of starlike functions are obtained as applications