Schwarzian norm estimates for some classes of analytic and harmonic mappings

Abstract

Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$. For $\beta >1$, let $\mathcal{N}(\beta )$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1+z{f}^{\prime \prime }(z)/f^{\prime }(z)\}<\beta$ for $z\in \mathbb{D}$. The main purpose of this paper is to establish sharp bounds for the pre-Schwarzian norm $\Vert P_f\Vert$ and Schwarzian norm $\Vert S_f\Vert$ for functions $f\in \mathcal{N}(\beta )$, parametrized by $f^{\prime \prime }(0)$, with special emphasis on the case $f^{\prime \prime }(0)=0$. In addition, sharp growth, distortion, and radius results (convexity and concavity) for $\mathcal{N}(\beta )$ are obtained. As an application, we determine the sharp pre-Schwarzian norm estimate for harmonic mappings $f=h+\bar{g}$ whose analytic part $h$ belongs to $\mathcal{N}(\beta )$.

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