Variability regions for the fourth derivative of bounded analytic functions

TL;DR

This paper derives a fourth-order Dieudonné's Lemma for bounded analytic functions and characterizes the variability region of the fourth derivative at a point, including extremal functions, in the unit disk.

Contribution

It establishes the fourth-order Dieudonné's Lemma and applies it to explicitly determine the variability region of the fourth derivative for bounded analytic functions with given conditions.

Findings

Derived the fourth-order Dieudonné's Lemma.

Determined the variability region for the fourth derivative.

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