On analytic functions related to Booth-lemniscate

Abstract

For $0\le \alpha\le 1 $, let $\mathcal{BS}(\alpha)$ be the class of all analytic functions in the unit disk $\mathbb{D}:=\{~z\in\mathbb{C}:|z|<1\}$ with normalization $f(0)=0$ and $f'(0)=1$ that satisfy the subordinate relation $zf'(z)/f(z)-1\prec z/(1-\alpha z^2)$ and $\mathcal{BK}(\alpha)$ be the class of all functions $f$ for which $zf' \in \mathcal{BS}(\alpha)$. In this article, we obtain a sharp estimate of the initial Taylor coefficients and logarithmic coefficients for functions in the classes $\mathcal{BS}(\alpha)$ and $\mathcal{BK}(\alpha)$. Further, we obtain the radius of convexity and study the pre-Schwarzian norm for the classes $\mathcal{BS}(\alpha)$ and $\mathcal{BK}(\alpha)$.