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 $\mathcal{G}(β)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]<1+β/2\}$, where $β>0$ and $\mathcal{F}(α)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]>α\}$, where $-1/2\le α\le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(β)$ and $\mathcal{F}(α)$.