Schwarzian norm estimates for some classes of analytic functions

Abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ normalized by $f(0)=0$, $f^{\prime }(0)=1$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $\mathcal{G}(\beta )=\{f\in \mathcal{A}:\mathrm{Re}[1+z{f}^{\prime \prime }(z)/f^{\prime }(z)]<1+\beta /2\}$, where $\beta >0$ and $\mathcal{F}(\alpha )=\{f\in \mathcal{A}:\mathrm{Re}[1+z{f}^{\prime \prime }(z)/f^{\prime }(z)]>\alpha \}$, where $-1/2\le \alpha \le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(\beta )$ and $\mathcal{F}(\alpha )$.