On analytic functions related to Booth-lemniscate

Abstract

For $0\le \hat{I}\pm \le 1$, let $\mathcal{BS}(\hat{I}\pm )$ 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 $f^{\prime }(0)=1$ that satisfy the subordinate relation $z{f}^{\prime }(z)/f(z)-1\prec z/(1-\hat{I}\pm z^2)$ and $\mathcal{B}\mathcal{K}(\hat{I}\pm )$ be the class of all functions $f$ for which $z{f}^{\prime }\in \mathcal{BS}(\hat{I}\pm )$. In this article, we obtain a sharp estimate of the initial Taylor coefficients and logarithmic coefficients for functions in the classes $\mathcal{BS}(\hat{I}\pm )$ and $\mathcal{B}\mathcal{K}(\hat{I}\pm )$. Further, we obtain the radius of convexity and study the pre-Schwarzian norm for the classes $\mathcal{BS}(\hat{I}\pm )$ and $\mathcal{B}\mathcal{K}(\hat{I}\pm )$.