Pre-Schwarzian and Schwarzian norm Estimates for Robertson class

Abstract: Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}={z\in\mathbb{C} : |z|<1}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-π/2<α<π/2$, let $\mathcal{S}α$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}{e^{iα}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)}>0$ for $z\in\mathbb{D}$. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass $\mathcal{S}α$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$.