Starlikeness for certain close-to-star functions

We find the radius of starlikeness of order $\alpha$, $0\leq \alpha<1$, of normalized analytic functions $f$ on the unit disk satisfying either $Reinsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(z);/ginsert ignore into journalissuearticles values(z););>0$ or $\left|insert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(z);/ginsert ignore into journalissuearticles values(z););-1\right|<1$ for some close-to-star function $g$ with $Reinsert ignore into journalissuearticles values(ginsert ignore into journalissuearticles values(z);/insert ignore into journalissuearticles values(z+z^2/2););>0$ as well as of the class of close-to-star functions $f$ satisfying $Reinsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(z);/insert ignore into journalissuearticles values(z+z^2/2););>0$. 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.