Sufficient conditions and radius problems for the Silverman class

Abstract

For $0<\alpha \le 1$ and $\lambda >0,$ let \begin{equation}\label{1} G_{λ,α}=\left\{f\in\mathcal{A}:\left|\dfrac{1-α+αzf''(z)/f'(z)}{zf'(z)/f(z)}-(1-α)\right|<λ, 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} Ω=\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 ${\mathcal{S}}^{\ast }$ and find radius estimates for different pairs involving these classes.