Starlikeness for certain close-to-star functions

Abstract

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