Variability regions for the second derivative of bounded analytic functions

Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $\mathrm{\mid }{w}_0\mathrm{\mid }<\mathrm{\mid }{z}_0\mathrm{\mid }$. Let ${\mathcal{H}}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$, and $$\mathcal{H}$_0 (z_0,w_0) = { f \in $\mathcal{H}$_0 : f(z_0) =w_0}$. In this paper, we explicitly determine the variability region of $f^{\mathrm{\prime }\mathrm{\prime }}(z_0)$ when $f$ ranges over $$\mathcal{H}$_0 (z_0,w_0)$. We also show a geometric view of our main result by Mathematica.