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

Abstract

We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_{\Omega }^j(z_0,c)=\{{\int }_0^{z_0}{z}^j(g(z)-g(0))dz:g(\mathbb{D})\subset \Omega ,(P^{-1}\circ g)(z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots \}$ for each fixed $z_0\in \mathbb{D}$, $j=-1,0,1,2,\dots$ and $c=(c_0,c_1,\dots ,c_n)\in {\mathbb{C}}^{n+1}$, when $\Omega ⊊\mathbb{C}$ 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 $\mathcal{C}\mathcal{V}(\Omega )$ be the class of analytic functions $f$ in $\mathbb{D}$ with $f(0)=f^{\prime }(0)-1=0$ satisfying $1+z{f}^{\prime \prime }(z)/f^{\prime }(z)\in \Omega$. As applications we determine variability regions of $\log f^{\prime }(z_0)$ when $f$ ranges over $\mathcal{C}\mathcal{V}(\Omega )$ with or without the conditions $f^{\prime \prime }(0)=\lambda$ and $f^{\prime \prime \prime }(0)=\mu$. Here $\lambda$ and $\mu$ are arbitrarily preassigned values. By choosing particular $\Omega$, we obtain the precise variability regions of $\log f^{\prime }(z_0)$ for other well-known subclasses of analytic and univalent functions.

arXiv comment13 pages