  By considering the polynomial function $\phi_{car}(z)=1+z+z^2/2,$ we define
the class $\Scar$ consisting of normalized analytic functions $f$ such that
$zf'/f$ is subordinate to $\phi_{car}$ in the unit disk. The inclusion
relations and various radii constants associated with the class $\Scar$ and its
connection with several well-known subclasses of starlike functions is
established. As an application, the obtained results are applied to derive the
properties of the partial sums and convolution.
