Some properties for \(\alpha\)-starlike functions with respect to \(k\)-symmetric points of complex order

In the present work, we introduce the subclass \(\mathcal{T}_{\gamma ,\alpha}^{k}(\varphi )\), of starlike functions with respect to \(k\)-symmetric points of complex order \(\gamma\) (\(\gamma \neq 0\)) in the open unit disc \(\vartriangle\). 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.