Starlikeness associated with certain strongly functions

Let κ be an analytic function in the open unit disc $$ in the complex plane with $$ . Then $$ if and only if $$ , where ≺ is a subordination relation and the function $$ , for $$ , maps $$ to a symmetric domain about the real axis in the right-half plane. This article finds the sharp radii of starlikeness for three classes of normalized analytic functions in the open unit disc. It also finds the solutions to first-order differential equations and utilizes these results together with the differential subordination developed by Miller and Mocanu to obtain the conditions so that some first-order differential relations associated with $$ are subordinate to certain Ma and Minda functions. Consequently, it provides sufficient conditions for the normalized analytic functions to belong to some established subclasses of starlike functions.