Sufficient Conditions and Radius Problems for the Silverman Class

For 0 < α ≤ 1 and λ > 0, let

\({G}_{\lambda ,\alpha }=\left\{f \in A: \left|\frac{1-\alpha +\alpha zf^{\prime\prime}\left(z\right)/{f}^{{^{\prime}}}\left(z\right)}{z{f}^{{^{\prime}}}\left(z\right)/f\left(z\right)}-\left(1-\alpha \right)\right|< \lambda , z \in {\mathbb{D}}\right\}. (0.1)\)

The general form of the Silverman class was introduced by Tuneski and Irmak [Int. J. Math. Math. Sci., 2006, Article ID 38089 (2006)]. Our differential-inequality formulation is based on several sufficient conditions for this class. Further, we consider a class Ω given by

\(\Omega =\left\{f\in A:\left|z{f{^{\prime}}}^{\left(z\right)}-f\left(z\right)\right|<\frac{1}{2},z\in {\mathbb{D}}\right\}. (0.2)\)

For these two classes, we establish inclusion relations involving some well-known subclasses of S^* and compute radius estimates featuring various pairings of these classes.