Starlikeness associated with certain strongly functions

Let κ be an analytic function in the open unit disc U in the complex plane with κ(0)=1. Then κ∈K(λ) if and only if κ(t)≺ϕ_λ(t), where ≺ is a subordination relation and the function ϕ_λ(t):=(1+t)^λ, for 0<λ<1, maps U 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 ϕ_λ 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.