Geometric subfamily of locally univalent functions, Blaschke products and quasidisk

TL;DR

This paper investigates a geometric family of functions characterized by a real part condition, revealing their properties, connections to Blaschke products, quasidisks, and establishing sharp bounds for their Schwarzian and pre-Schwarzian derivatives.

Contribution

It introduces the family () with new geometric and analytic properties, including sharp bounds and connections to quasidisks and Blaschke products.

Findings

The image of functions in () are quasidisks.

Established the exact Schwarzian norm  Sf  = 2(2-).

Derived sharp bounds for the pre-Schwarzian norm of harmonic mappings.

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 $\mathcal{F}(\alpha)$.