Starlikeness of Analytic Functions with Subordinate Ratios

Let be a nonvanishing analytic function in the open unit disc with . Consider the class consisting of normalized analytic functions whose ratios , , and are each subordinate to for some analytic functions and . The radius of starlikeness of order is obtained for this class when is chosen to be either or . Further, starlikeness radii are also obtained for each of these two classes, which include the radius of Janowski starlikeness, and the radius of parabolic starlikeness.