Strong differential subordination and applications to univalency conditions

Abstract

For the Briot-Bouquet differential equations of the fo-rm given in \cite{1} \begin{equation*} u(z)+\frac{zu^{\prime }(z)}{z\dfrac{f^{\prime }(z)}{f(z)}\left[ \alpha u(z)+\beta \right] }=g(z), \end{equation*}% we can reduce them to \begin{equation*} v(z)+F(z)\frac{v^{\prime }(z)}{v(z)}=h(z), \end{equation*}% where $v(z)=\alpha u(z)+\beta ,\ h(z)=\alpha g(z)+\beta $ \ and $% F(z)=f(z)/f^{\prime }(z).$ In this paper we are going to give conditions in order that if $u$ and $v$ satisfy, respectively, the equations% \begin{align} u(z)+F(z)\frac{u^{\prime }(z)}{u(z)} =&\ h(z), \label{1} \\ v(z)+G(z)\frac{v^{\prime }(z)}{v(z)} =&\ g(z) \notag \end{align}% with certain conditions on the functions $F$ and $G$ applying the concept of strong subordination $g\prec \prec h$ given in \cite{2} by the author, implies that $v\prec u$, where $\prec $ indicates subordination.