Convolution conditions for bounded $\alpha$-starlike functions of complex order

Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $\mathbb{C}$ with the normalization $f(0)=f^{{}^{\prime }}(0)-1=0$. We introduce a subclass $S_M^{\ast }(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_M^{\ast }(n,\alpha ,b)$ ($n\ge 0$) related to $S_M^{\ast }(\alpha ,b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f\in A$ to belong to the class $S_M^{\ast }(\alpha ,b)$. Several properties of the class $S_M^{\ast }(n,\alpha ,b)$ are investigated.