Region of variability for certain subclass of univalent functions

TL;DR

This paper determines the region of variability for the logarithmic derivative of a subclass of univalent functions defined on the unit disk, expanding understanding of their geometric properties.

Contribution

It explicitly characterizes the variability region of \,\log f'_{\alpha}(z_0)\, for functions in a specific subclass of univalent functions with given initial conditions.

Findings

Derived the region of variability for \log f'_{\alpha}(z_0).

Extended known results to a subclass defined by a real part condition.

Provided explicit bounds for the logarithmic derivative in the class.

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 ${\mathcal F}(\lambda):=\{f_{\alpha} \in \mathcal{F}: f_{\alpha}(0)=0, f'_{\alpha}(0)=1 \, \mbox{and} \, f"_{\alpha}(0)=2\lambda (1-\beta) \,\,\, \mbox{for} \,\, 0\leq \lambda \leq 1\}$.