An application of Schur algorithm to variability regions of certain analytic functions-II

TL;DR

This paper extends previous work on variability regions of certain analytic functions, applying the Schur algorithm to starlike domains and deriving precise variability regions for subclasses of univalent functions.

Contribution

It generalizes earlier results to starlike domains and determines variability regions of log derivatives for various subclasses of analytic functions.

Findings

Variability regions are characterized for functions with starlike domains.

Explicit regions are obtained for subclasses with fixed derivatives at zero.

Results include variability regions for well-known univalent function classes.

Abstract

We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_\Omega^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g({\mathbb D}) \subset \Omega, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$ for each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in \mathbb{C}^{n+1}$, when $\Omega\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of the unit disk ${\mathbb D}$ onto $\Omega$. In the present article, we first show that in the case $n=0$, $j=-1$ and $c=0$, the result obtained in \cite{Ali-Vasudevarao-Yanagihara-2018} still holds when one assumes only that $\Omega$ is starlike with respect to $P(0)$. Let $\mathcal{CV}(\Omega)$ be the class of analytic functions $f$ in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega}$. As applications we determine variability regions of $\log f'(z_0)$ when $f$ ranges over $\mathcal{CV}(\Omega)$ with or without the conditions $f''(0)= \lambda$ and $f'''(0)= \mu$. Here $\lambda$ and $\mu$ are arbitrarily preassigned values. By choosing particular $\Omega$, we obtain the precise variability regions of $\log f'(z_0)$ for other well-known subclasses of analytic and univalent functions.