Variability regions for the fourth 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 fourth-order Dieudonné's Lemma and apply it to determine the variability region $\{f^{(4)}(z_0):f\in {\mathcal{H}}_0,f(z_0)=w_0,f^{\prime }(z_0)=w_1,f^{\prime \prime }(z_0)=w_2\}$ for given $z_0,w_0,w_1,w_2$ and give the form of all the extremal functions.

arXiv comment12 pages