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 - \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 $f\in \mathcal{F}(\alpha )$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f\in \mathcal{F}(\alpha )$, then $\Vert S_f\Vert =2\alpha (2-\alpha )$. Moreover, we establish the sharp estimate $\Vert P_f\Vert \le 2\alpha +1$ 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 $\mathcal{F}(\alpha )$.

20 pages, 3 figurs. arXiv admin note: text overlap with arXiv:2407.14922 by other authors