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 $\mathcal{C}_θ(γ)$ 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 $\mathcal{C}_θ(γ)$. We establish sharp estimates for the Schwarzian norm $\|S_f\|$ of functions $f$ in the class $\mathcal{C}_θ(γ)$ and derive univalence criteria using both pre-Schwarzian and Schwarzian norm estimates. We also introduce a corresponding harmonic class $\mathcal{HC}_θ(γ)$ consisting of mappings $f = h+\overline{g}$ with $h\in\mathcal{C}_θ(γ)$ and dilatation $ω=g'/h'\in\mathrm{Aut}(\mathbb{D})$. For this harmonic class, we derive bounds for both the pre-Schwarzian and Schwarzian norms, including sharp results in special cases.

Figures (1)

• Figure 1: Image of $\mathbb{D}$ under the mapping $g(z)$ for different values of $\theta$ and $\gamma$

Theorems & Definitions (18)

• Lemma 1.1
• Lemma 1.2
• Theorem 2.1
• proof
• Remark 2.1
• Theorem 2.2
• proof
• Corollary 2.1
• Corollary 2.2
• Remark 2.2