Inclusion relations and radius problems for a subclass of starlike functions

By considering the polynomial function ${\varphi }_{car}(z)=1+z+z^2\mathrm{/}2,$ we define the class $\text{Scar}$ consisting of normalized analytic functions $f$ such that $z{f}^{\mathrm{\prime }}\mathrm{/}f$ is subordinate to ${\varphi }_{car}$ in the unit disk. The inclusion relations and various radii constants associated with the class $\text{Scar}$ 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.