Variability regions for the second derivative of bounded analytic functions

Abstract: Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$. 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 $f''(z_0)$ when $f$ ranges over $\mathcal{H}_0 (z_0,w_0)$. We also show a geometric view of our main result by Mathematica.