  Let $f$ and $g$ be analytic functions on the open unit disk of the complex
plane with $f/g$ belonging to the class $\mathcal{P} $ of functions with
positive real part consisting of functions $p$ with $p(0)=1$ and
$\operatorname{Re} p(z)>0$ or to its subclass consisting of functions $p$ with
$|p(z)-1|<1$. We obtain the sharp radius constants for the function $f$ to be
starlike of order $\alpha$, parabolic starlike, etc. when $g/k\in\mathcal{P}$
where $k$ denotes the Koebe function defined by $k(z)=z/(1-z)^2$.
