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 $-\pi /2<\alpha <\pi /2$, let ${\mathcal{S}}_{\alpha }$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{i\alpha }\left(1+z{f}^{\prime \prime }(z)/f^{\prime }(z)\right)\}>0$ 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 ${\mathcal{S}}_{\alpha }$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime }(0)$.

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