The Booth Lemniscate Starlikeness Radius for Janowski Starlike Functions

TL;DR

This paper determines the largest radii for classes of functions related to Booth lemniscate mappings, especially for Janowski starlike functions, by analyzing the geometric properties of the associated domain.

Contribution

It introduces new radius bounds for Janowski starlike functions within the Booth lemniscate framework, extending known results in geometric function theory.

Findings

Largest radius for starlike functions of order β

Radius for Janowski starlike functions

Disc containment within Booth lemniscate domain

Abstract

The function $G_\alpha(z)=1+ z/(1-\alpha z^2)$, \, $0\leq \alpha <1$, maps the open unit disc $\mathbb{D}$ onto the interior of a domain known as the Booth lemniscate. Associated with this function $G_\alpha$ is the recently introduced class $\mathcal{BS}(\alpha)$ consisting of normalized analytic functions $f$ on $\mathbb{D}$ satisfying the subordination $zf'(z)/f(z) \prec G_\alpha(z)$. Of interest is its connection with known classes $\mathcal{M}$ of functions in the sense $g(z)=(1/r)f(rz)$ belongs to $\mathcal{BS}(\alpha)$ for some $r$ in $(0,1)$ and all $f \in \mathcal{M}$. We find the largest radius $r$ for different classes $\mathcal{M}$, particularly when $\mathcal{M}$ is the class of starlike functions of order $\beta$, or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disc contained in $G_\alpha(\mathbb{D})$ and centered at a certain point $a \in \mathbb{R}$.