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

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