  Let $h$ be a non-vanishing analytic function in the open unit disc with
$h(0)=1$. Consider the class consisting of normalized analytic functions $f$
whose ratios $f(z)/g(z)$, $g(z)/z p(z)$, and $p(z)$ are each subordinate to $h$
for some analytic functions $g$ and $p$. The radius of starlikeness is obtained
for this class when $h$ is chosen to be either $h(z)=\sqrt{1+z}$ or $h(z)=e^z$.
Further $\mathcal{G}$-radius is also obtained for each of these two classes
when $\mathcal{G}$ is a particular widely studied subclass of starlike
functions. These include $\mathcal{G}$ consisting of the Janowski starlike
functions, and functions which are parabolic starlike.
