Variability regions for the third derivative of bounded analytic functions

Abstract

Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0|<|z_0|$, and ${\mathcal{H}}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$. In this paper, we establish the third order Dieudonné Lemma, and apply it to explicitly determine the variability region $\{f^{\prime \prime \prime }(z_0):f\in {\mathcal{H}}_0,f(z_0)=w_0,f^{\prime }(z_0)=w_1\}$ for given $z_0,w_0,w_1$ and give the form of all the extremal functions.

10 pages