On analytic functions related to Booth-lemniscate

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