Schwarzian Norm Estimate for Functions in Generalized Robertson Class

Abstract

Let $\mathcal{A}$ be the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ with the normalized conditions $f(0)=0$, $f^{\prime }(0)=1$. For $-\pi /2<\alpha <\pi /2$ and $0\le \beta <1$, let ${\mathcal{S}}_{\alpha }(\beta )$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $$\mathrm{Re}\left\{e^{i\alpha }\left(1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\right)\right\}>\beta \cos \alpha \text{for}z\in \mathbb{D}.$$ In the present study, we will compute the sharp estimate of the pre-Schwarzian and Schwarzian norms for functions in ${\mathcal{S}}_{\alpha }(\beta )$.

arXiv admin note: substantial text overlap with arXiv:2307.08976