Geometric subfamily of locally univalent functions, Blaschke products and quasidisk
Molla Basir Ahamed, Rajesh Hossain
TL;DR
This work introduces the local-univalence class $\mathcal{F}(\alpha)$, defined by $\operatorname{Re}\left(1+\frac{z f''(z)}{f'(z)}\right)>1-\frac{\alpha}{2}$, and investigates its geometric and analytic structure. The authors establish subordination and Blaschke-product representations for $f'$ and prove that $f(\mathbb{D})$ is a quasidisk, with sharp Schwarzian and pre-Schwarzian estimates: $\|S_f\|\le 2\alpha(2-\alpha)$ and, for harmonic mappings with fixed analytic parts in $\mathcal{F}(\alpha)$, $\|P_f\|\le 2\alpha+1$. They also obtain a sharp Bloch-constant bound for the associated harmonic class $\mathcal{F}_{\mathcal{H}}(\alpha)$ and analyze univalent harmonic mappings whose analytic parts lie in $\mathcal{F}(\alpha)$, including conditions ensuring univalence and quasidisk targets. Collectively, the results connect subordination, Blaschke products, Schwarzian theory, and quasiconformal geometry within this subfamily, providing sharp bounds and explicit extremals that illuminate the geometry of $\mathcal{F}(\alpha)$ and its harmonic companions.
Abstract
In this article, we consider the family $\mathcal{F}(α)$ defined for $α\in (0, 3]$ by \begin{align*} {\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \fracα{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 $f \in \mathcal{F}(α)$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f \in \mathcal{F}(α)$, then $\|S_f\| = 2α(2-α)$. Moreover, we establish the sharp estimate $\|P_{f}\| \leq 2α+1$ for the pre-Schwarzian derivative of harmonic mappings $f = h + \bar{g} \in \mathcal{F}_{\mathcal{H}}(α)$, where the analytic part $h$ belongs to $\mathcal{F}(α)$.