On Booth lemniscate and starlike functions

Abstract

Let $$\Delta $$ be the open unit disk in the complex plane and $$\mathcal {A}$$ be the class of normalized analytic functions in $$\Delta $$ . In this paper, we introduce and study the class $$\begin{aligned} \mathcal {BS}(\alpha ):=\left\{ f\in \mathcal {A}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{1-\alpha z^2}, \, z\in \Delta \right\} , \end{aligned}$$ where $$0\le \alpha \le 1$$ and $$\prec $$ is the subordination relation. Some properties of this class like differential subordination, coefficient estimates and Fekete–Szego inequality associated with the k-th root transform are considered.