Starlikeness of a product of starlike functions with non-vanishing polynomials

For a function f starlike of order $$\alpha $$ , $$0\leqslant \alpha <1$$ non-constant polynomial Q degree n which is non-vanishing in the unit disc $${\mathbb {D}}$$ and $$\beta >0$$ we consider $$F:{\mathbb {D}}\rightarrow {\mathbb {C}}$$ defined by $$F(z)=f(z) (Q(z))^{\beta /n}$$ find largest value $$r\in (0,1]$$ such that $$r^{-1} F(rz)$$ lies various known subclasses functions as class $$\uplambda classes associated with exponential function, cardioid, rational nephroid domain modified sigmoid function. Our radii results are sharp. We also discuss correlation special cases.