Region of variability for certain subclass of univalent functions

Abstract

Let $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ be the unit disk. For $0<\alpha <1$, let $f_{\alpha }(z)=z/(1-z^{\alpha })$ for $z\in \mathbb{D}$. We consider the class $\mathcal{F}$ of analytic functions $f_{\alpha }$ which satisfy $\Re \left(1+zf{"}_{\alpha }(z)/f_{\alpha }^{\prime }(z)\right)>\beta$ for $0<\beta <1$. In this paper, we determine the region of variability of $\log f_{\alpha }^{\prime }(z_0)$ for fixed $z_0\in \mathbb{D}$ when $f$ varies over the class ${\mathcal F}(λ):=\{f_α \in \mathcal{F}: f_α(0)=0, f'_α(0)=1 \, \mbox{and} \, f"_α(0)=2λ(1-β) \,\,\, \mbox{for} \,\, 0\leq λ\leq 1\}$.

19 pages, 10 figures