Briot-Bouquet differential subordination and Bernardi's integral operator

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