Further results for starlike functions related with Booth lemniscate

TL;DR

This paper explores a subclass of starlike functions related to the Booth lemniscate, analyzing geometric properties, radii of starlikeness, image domains, and bounds for the real part of the functions.

Contribution

It introduces and investigates geometric properties of the subclass () of starlike functions connected with the Booth lemniscate, including radii and bounds.

Findings

Determined the radius of starlikeness of order b3.

Described the image of the unit disk under functions in ().

Provided bounds for f(z)/zf(z).

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.