Ruscheweyh derivative and strongly starlike functions

Abstract

Let $A$ denote the class of analytic functions $f(z)$ defined in the unit disc satisfying the condition $f(O)=f^{\prime}(0)-1=0$. Let $\overline{S}^{*}(\beta, \gamma)$ be the class of strongly starlike functions of order $\beta$ and type $\gamma$, and let $\overline{C}(\beta, \gamma)$ denote the class of strongly convex functions of order $\beta$ and type $\gamma$. Certain new classes $\overline{S}_{\alpha}^{*}(\beta, \gamma)$ and $\overline{C}_{\alpha}(\beta, \gamma)$ are introduced by virtue of Ruscheweyh derivative and some properties of $\overline{S}_{\alpha}^{*}(\beta, \gamma)$ and $\overline{C}_{\alpha}(\beta, \gamma)$ are discussed.