Schwarzian norm estimates for some classes of analytic functions

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