Hadamard product of holomorphic mappings associated with the conic shaped domain

Abstract

We define certain subclasses $\delta-\mathcal{UM}(\ell,\eta_{1},\eta_{2})$ and $\delta-\mathcal{UM}_{\Im}(\ell,\eta_{1},\eta_{2})$ of holomorphic mappings involving some differential inequalities. These functions are actually generalizations of some basic families of starlike and convex mappings. We study sufficient conditions for $f\in \delta-\mathcal{UM}(\ell,\eta_{1}% ,\eta_{2}).$ We also discuss the characterization for $f\in \delta -\mathcal{UM}_{\Im}(\ell,\eta_{1},\eta_{2})$ along with the coefficient bounds and other problems. Using certain conditions for functions in the class $\delta-\mathcal{UM}(\ell,\eta_{1},\eta_{2}),$ we also define another class $\delta-\mathcal{UM}^{\ast}(\ell,\eta_{1},\eta_{2})$ and study some subordination related result.