Further results for starlike functions related with Booth lemniscate

Abstract

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.