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 $f^{\prime \prime }(z_0)$ when $f$ ranges over ${\mathcal{H}}_0(z_0,w_0)$. We also show a geometric view of our main result by Mathematica.

11 pages, 4 figures