The Booth Lemniscate Starlikeness Radius for Janowski Starlike Functions

Abstract

The function $G_{\alpha }(z)=1+z/(1-\alpha z^2)$, \, $0\le \alpha <1$, 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 $z{f}^{\prime }(z)/f(z)\prec G_{\alpha }(z)$. 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 $a\in \mathbb{R}$.