Pre-Schwarzian and Schwarzian norm Estimates for Robertson class

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$, normalized by $f(0)=0$ and $f^{\mathrm{\prime }}(0)=1$. For $-\pi \mathrm{/}2<\alpha <\pi \mathrm{/}2$, let ${\mathcal{S}}_{\alpha }$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{R}\mathrm{e}\{e^{i\alpha }(1+z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z))\}>0$ 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 ${\mathcal{S}}_{\alpha }$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\mathrm{\prime }\mathrm{\prime }}(0)$.