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 $f(0)=f^{\prime }(0)-1=0$. In this paper, we introduce a new subclass ${\mathcal{C}}_{\theta }(\gamma )$ of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation \[ \textrm{Re}\left(e^{iθ}\left(1+\frac{zf''(z)}{f'(z)}\right)\right)<\left(1+\fracγ{2}\right)\cosθ,~ z\in\mathbb{D},~ γ>0, ~\text{and}~|θ|<\fracπ{2},\] 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 $\Vert S_f\Vert$ 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 ${\mathcal{H}\mathcal{C}}_{\theta }(\gamma )$ consisting of mappings $f=h+\overline{g}$ with $h\in {\mathcal{C}}_{\theta }(\gamma )$ and dilatation $\omega =g^{\prime }/h^{\prime }\in \mathrm{Aut}(\mathbb{D})$. For this harmonic class, we derive bounds for both the pre-Schwarzian and Schwarzian norms, including sharp results in special cases.

15 pages