Region of variability for certain subclass of univalent functions

Let $\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ be the unit disk. For $0<\alpha <1$, let $f_{\alpha }(z)=z\mathrm{/}(1-z^{\alpha })$ for $z\in \mathbb{D}$. We consider the class $\mathcal{F}$ of analytic functions $f_{\alpha }$ which satisfy $\mathrm{\Re }(1+zf{\mathrm{"}}_{\alpha }(z)\mathrm{/}{f}_{\alpha }^{\mathrm{\prime }}(z))>\beta$ for $0<\beta <1$. In this paper, we determine the region of variability of $\log f_{\alpha }^{\mathrm{\prime }}(z_0)$ for fixed $z_0\in \mathbb{D}$ when $f$ varies over the class $\mathcal{F}(\lambda ):=\{f_{\alpha }\in \mathcal{F}:f_{\alpha }(0)=0,f_{\alpha }^{\mathrm{\prime }}(0)=1andf{\mathrm{"}}_{\alpha }(0)=2\lambda (1-\beta )for0\le \lambda \le 1\}$.