  In this paper we investigate an interesting subclass $\mathcal{BS}(\alpha)$
($0\leq \alpha<1$) of starlike functions in the unit disk $\Delta$. The class
$\mathcal{BS}(\alpha)$ was introduced by Kargar et al. [R. Kargar, A. Ebadian
and J. Sok\'o{\l}, {\it On Booth lemniscate and starlike functions}, Anal.
Math. Phys. (2017) DOI: 10.1007/s13324-017-0187-3] which is strongly related to
the Booth lemniscate. Some geometric properties of this class of analytic
functions including, radius of starlikeness of order $\gamma$
($0\leq\gamma<1$), the image of $f(\{z:|z|<r\})$ when $f\in
\mathcal{BS}(\alpha)$, an special example and estimate of bounds for ${\rm
Re}\{f(z)/z\}$ are studied.
