Region of variability for certain subclass of univalent functions (2209.10424v1)

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 $\Re \left(1+zf"{\alpha}(z)/f'{\alpha}(z)\right) > \beta$ for $0<\beta<1$. In this paper, we determine the region of variability of $\log f'{\alpha}(z_0)$ for fixed $z{0} \in \mathbb{D}$ when $f$ varies over the class ${\mathcal F}(\lambda):={f_{\alpha} \in \mathcal{F}: f_{\alpha}(0)=0, f'{\alpha}(0)=1 \, \mbox{and} \, f"{\alpha}(0)=2\lambda (1-\beta) \,\,\, \mbox{for} \,\, 0\leq \lambda \leq 1}$.