a )$ of close-to-convex functions of order $\alpha $ and type $\delta $.  $(ii)$ If we let $\delta =0,\;\beta \rightarrow \infty $ in Definition $\ref{dfn0}$, then the class $\mathcal{S}_{g}\left( \alpha ,\beta \right) $ reduces to the class $\mathcal{C}(\alpha )$ of close-to-convex functions of order $\alpha $