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 $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{} < 1\}$. For $\beta > 1$, let $\mathcal{N}(\beta)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\} < \beta$ 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 $f''(0)$, with special emphasis on the case $f''(0) = 0$. 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 $\mathcal{N}(\beta)$.