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Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f\left(0\right)=f^{{}^{\text{'}}}\left(0\right)-1=0$. We introduce a subclass $S_M^{*}(\alpha ,b)$ of $A$, which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_M^{*}(n,\alpha ,b)$ ($n\ge 0$) related to $S_M^{*}(\alpha ,b)$ 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 $S_M^{*}(\alpha ,b)$. Several properties of the class $S_M^{*}(n,\alpha ,b)$ are investigated.

In the present work, we introduce the subclass ${𝒯}_{\gamma ,\alpha }^k\left(\varphi \right)$, of starlike functions with respect to $k$-symmetric points of complex order $\gamma$ ($\gamma \ne 0$) in the open unit disc $$. Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.