Pre-Schwarzian and Schwarzian norm estimates for certain classes of analytic and harmonic mappings

Abstract

Let $\mathcal{A}$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $f(0)=f'(0)-1=0$. In this paper, we introduce a new subclass $\mathcal{C}_\theta(\gamma)$ of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation \[ \textrm{Re}\left(e^{i\theta}\left(1+\frac{zf''(z)}{f'(z)}\right)\right)<\left(1+\frac{\gamma}{2}\right)\cos\theta,~ z\in\mathbb{D},~ \gamma>0, ~\text{and}~|\theta|<\frac{\pi}{2},\] and investigate the Schwarzian derivative and Schwarzian norm for functions $f$ belonging to the class $\mathcal{C}_\theta(\gamma)$. We establish sharp estimates for the Schwarzian norm $\|S_f\|$ of functions $f$ in the class $\mathcal{C}_{\theta}(\gamma)$ and derive univalence criteria using both pre-Schwarzian and Schwarzian norm estimates. We also introduce a corresponding harmonic class $\mathcal{HC}_{\theta}(\gamma)$ consisting of mappings $f = h+\overline{g}$ with $h\in\mathcal{C}_{\theta}(\gamma)$ and dilatation $\omega=g'/h'\in\mathrm{Aut}(\mathbb{D})$. For this harmonic class, we derive bounds for both the pre-Schwarzian and Schwarzian norms, including sharp results in special cases.