Certain properties of Bazilevi\(\breve{c}\) type univalent class defined through subordination

In the present paper with the aid of subordination, the authors introduce two subclasses of analytic functions denoted by \({\mathcal {S}}_{\alpha , \beta }(\lambda )~~(\alpha ,~\beta ,~ \lambda \in {\mathbb {R}},~\alpha <1, \beta >1, \lambda \ge 0)\) ${\mathcal{S}}_{\alpha ,\beta }(\lambda )(\alpha ,\beta ,\lambda \in \mathbb{R},\alpha <1,\beta >1,\lambda \ge 0)$ and \({\mathcal {G}}(\lambda )\) $\mathcal{G}(\lambda )$ defined in the open unit disk \({\mathbb {D}}:=\{z \in {\mathbb {C}}:|z|<1\}\) $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ . These subclasses are defined through a certain univalent function \({\mathcal {S}}_{\alpha , \beta }\) ${\mathcal{S}}_{\alpha ,\beta }$ and the generating function of the Gregory coefficients \({\mathcal {G}}(\lambda )\) $\mathcal{G}(\lambda )$ . We determine upper bounds of the initial coefficients, Fekete–Szeg \(\ddot{o}\) $\ddot{o}$ functional, Hankel determinant of second order, logarithmic coefficients and inverse coefficients of the functions belongs to these subclasses. Some of the corollaries of the main results are also pointed out.