Schwarzian Norm Estimate for Functions in Robertson Class

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 $f(0)=0$, $f^{\prime }(0)=1$. For $-\pi /2<\alpha <\pi /2$, let ${\mathcal{S}}_{\alpha }$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{i\alpha }(1+z{f}^{\prime \prime }(z)/f^{\prime }(z))\}>0$ for $z\in \mathbb{D}$. In the present article, we determine the sharp estimate of the pre-Schwarzian and Schwarzian norms for functions in the class ${\mathcal{S}}_{\alpha }$.