Sharp radius of concavity for certain classes of analytic functions

Let $\mathcal{A}$ be the class of all analytic functions $f$ defined on the open unit disk $\mathbb{D}$ with the normalization $f(0)=0=f^{\mathrm{\prime }}(0)-1$. This paper examines the radius of concavity for various subclasses of $\mathcal{A}$, namely ${\mathcal{S}}_0^{(n)}$, $\mathcal{K}(\mathcal{\alpha },\mathcal{\beta })$, $\widetilde{{\mathcal{S}}^{\ast }}(\mathcal{\beta })$, and ${\mathcal{S}}^{\ast }(\alpha )$. It also presents results for various classes of analytic functions on the unit disk. All the radii are best possible.