Norm Estimates of the Pre-Schwarzian Derivatives for Functions with Conic-like Domains

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in U, where U is the open unit disk. pre-Schwarzian as well popular tools for studying geometric properties mappings. These can also be used to obtain either necessary or sufficient conditions univalence a function f. Because computational difficulty, norm has received more attention than norm. It applications theory hypergeometric functions, conformal mappings, Teichmüller spaces, and univalent functions. In this paper, we find sharp estimates certain subfamilies involving some conic-like image domains. results may extended families strongly starlike, convex, with symmetric conjugate points.