Starlikeness for Certain Close-to-Star Functions

We find the radius of starlikeness of order $\alpha$, $0\le \alpha <1$, of normalized analytic functions $f$ on the unit disk satisfying either $\mathrm{Re}(f(z)\mathrm{/}g(z))>0$ or $\mid (f(z)\mathrm{/}g(z))-1\mid <1$ for some close-to-star function $g$ with $\mathrm{Re}(g(z)\mathrm{/}(z+z^2\mathrm{/}2))>0$ as well as of the class of close-to-star functions $f$ satisfying $\mathrm{Re}(f(z)\mathrm{/}(z+z^2\mathrm{/}2))>0$. Several other radii such as radius of univalence and parabolic starlikeness are shown to be the same as the radius of starlikeness of appropriate order.