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

Let $\mathcal{A}$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ such that $f(0)=f^{\mathrm{\prime }}(0)-1=0$. In this paper, we introduce a new subclass ${\mathcal{C}}_{\theta }(\gamma )$ of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $$\text{Re}\left(e^{i\theta }(1+\frac{z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)}{f^{\mathrm{\prime }}(z)})\right)<(1+\frac{\gamma }{2})\cos \theta ,z\in \mathbb{D},\gamma >0,\text{ and }\mathrm{\mid }\theta \mathrm{\mid }<\frac{\pi }{2},$$ 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 $\mathrm{\parallel }{S}_f\mathrm{\parallel }$ 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 $f=h+\bar{g}$ with $h\in {\mathcal{C}}_{\theta }(\gamma )$ and dilatation $\omega =g^{\mathrm{\prime }}\mathrm{/}{h}^{\mathrm{\prime }}\in \mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{D})$. For this harmonic class, we derive bounds for both the pre-Schwarzian and Schwarzian norms, including sharp results in special cases.