Region of variability for spiral-like functions with respect to a boundary point
Volume 116 / 2009
Abstract
For $\mu\in\mathbb C$ such that ${\rm Re\,}\mu>0$ let ${\mathcal F}_{\mu} $ denote the class of all non-vanishing analytic functions $f$ in the unit disk $\mathbb{D}$ with $f(0)=1$ and $$ {\rm Re} \bigg(\frac{2\pi}{\mu}\, \frac{zf'(z)}{f(z)}+ \frac{1+z}{1-z}\bigg ) >0 \quad\ \hbox{in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk, $a\in\mathbb{C}$ with $|a|\leq 1$ and $\lambda\in\overline{\mathbb{D}}$, we shall determine the region of variability $V(z_0,\lambda)$ for $\log f(z_0)$ when $f$ ranges over the class \begin{multline*} \mathcal{F}_{\mu}(\lambda) = \biggl\{ f\in{\mathcal F}_{\mu} : f'(0)=\frac{\mu}{\pi}(\lambda-1) \hbox{ and}\\ f' '(0)=\frac{\mu}{\pi}\biggl(a(1-|\lambda|^2)+\frac{\mu}{\pi} (\lambda-1)^2-(1-{\lambda}^2)\biggr)\biggr\}.\end{multline*} In the final section we graphically illustrate the region of variability for several sets of parameters.
