Starlikeness of certain non-univalent functions

We consider three classes of functions defined using the class P of all analytic functions p(z) = 1+ cz + . . . on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with f/g is an element of P and g/(zp) is an element of P for some normalized analytic function g and p is an element of P. The second class is defined by replacing the condition f/g is an element of P by vertical bar(f/g) - 1 vertical bar < 1 while the other class consists of normalized analytic functions f with f/(zp) is an element of P for some p is an element of P. We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order alpha, parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.