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