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 $f(0)=0$, $f'(0)=1$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $\mathcal{G}(\beta)={f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]<1+\beta/2}$, where $\beta>0$ and $\mathcal{F}(\alpha)={f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]>\alpha}$, where $-1/2\le \alpha\le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(\beta)$ and $\mathcal{F}(\alpha)$.