Geometric subfamily of locally univalent functions, Blaschke products and quasidisk

Abstract

In this article, we consider the family $\mathcal{F}(\alpha)$ defined for $\alpha \in (0, 3]$ by \begin{align*} {\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \frac{\alpha}{2} \quad \text{for } z \in \mathbb{D}. \end{align*} Our primary objective is to show that this family possesses significant geometric and analytic properties, including connections with Blaschke products and the Schwarzian derivative, as well as its sharp bounds. Furthermore, we prove that if $f \in \mathcal{F}(\alpha)$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f \in \mathcal{F}(\alpha)$, then $\|S_f\| = 2\alpha(2-\alpha)$. Moreover, we establish the sharp estimate $\|P_{f}\| \leq 2\alpha+1$ for the pre-Schwarzian derivative of harmonic mappings $f = h + \bar{g} \in \mathcal{F}_{\mathcal{H}}(\alpha)$, where the analytic part $h$ belongs to $\mathcal{F}(\alpha)$.