Sufficient conditions and radius problems for the Silverman class

Abstract

For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}:\left|\dfrac{1-\alpha+\alpha zf''(z)/f'(z)}{zf'(z)/f(z)}-(1-\alpha)\right|<\lambda, z\in\mathbb{D}\right\}, \end{equation} the general form of Silverman class introduced by Tuneski and Irnak. For this class we derive some sufficient conditions in the form of differential inequalities. Further, we consider the class $\Omega,$ given by \begin{equation}\label{omega} \Omega=\left\{f\in\mathcal{A}:|zf'(z)-f(z)|<\dfrac{1}{2},\;z\in\mathbb{D}\right\}. \end{equation} For the above two classes, we establish inclusion relations involving some other well known subclasses of $\mathcal{S}^*$ and find radius estimates for different pairs involving these classes.