Starlikeness of certain non-univalent functions

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