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 $f(0)=0=f^{\prime}(0)-1$. This paper examines the radius of concavity for various subclasses of $\mathcal{A}$, namely $\mathcal{S}_0^{(n)}$, $\mathcal{K(\alpha,\beta)}$, $\mathcal{\tilde{S^*}(\beta)}$, and $\mathcal{S}^*(\alpha)$. It also presents results for various classes of analytic functions on the unit disk. All the radii are best possible.