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 disk $\mathbb {D}$ 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 disk contained in $G_\alpha (\mathbb {D})$ and centered at a certain point $a \in \mathbb {R}$.