Region of Variability for Some Subclasses of Univalent Functions

Abstract

Let $\fancyscript{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb {D}$ with $f(0)=0$ and $f'(0)=1$. Let $\fancyscript{S}$ denote the class of univalent functions in $\fancyscript{A}$. Let $\widetilde{\fancyscript{F}}$ (for example, class of starlike, convex, close-to-convex, spiralike, etc.) be any arbitrary subfamily of $\fancyscript{S}$ and $z_0\in \mathbb {D}$ then upper and lower estimates of $|f(z_0)|$, $|f'(z_0)|$ and $\mathrm{Arg} f'(z_0)$ for all $f\in \widetilde{\fancyscript{F}}$ are respectively called a growth theorem, a distortion theorem and a rotation theorem at $z_0$ for $\widetilde{\fancyscript{F}}$. These estimates deal only with absolute values of $f(z_0)$ and $f'(z_0)$ or with the argument of $f'(z_0)$. The aim of this paper is to give a survey on regions of variability of $f(z_0)$ or $f'(z_0)$ or $\log f'(z_0)$ when $f$ ranges over some well-known subclasses of $\fancyscript{S}$. As a consequence, we present the sharp Pre-Schwarzian norm and Block semi-norm for some of the subclasses of $\fancyscript{S}$. We also graphically illustrate the region of variability for several sets of parameters.

The author thank SRIC, IIT Kharagpur (ref. IIT/SRIC/MA/SUA/2012-13/144) for the financial support.