Region of variability for certain subclass of univalent functions

Jnana Preeti Parlapalli; Vasudevarao Allu

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arXiv · 2209.10424

Region of variability for certain subclass of univalent functions

Jnana Preeti Parlapalli·Vasudevarao Allu

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Abstract

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

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Jnana Preeti Parlapalli, Vasudevarao Allu. 2022-09-21. Region of variability for certain subclass of univalent functions. https://arxiv.org/abs/2209.10424

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