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.
Categories: math.CV