Geometric subfamily of functions convex in some direction and Blaschke products

TL;DR

This paper studies a family of locally univalent analytic functions in the unit disk, exploring their properties related to Blaschke products, Schwarzian derivatives, and univalent harmonic mappings under certain convexity conditions.

Contribution

It introduces and analyzes a new class of functions with specific convexity properties, linking them to Blaschke products and harmonic mappings, and explores their elegant mathematical properties.

Findings

The family exhibits connections to Blaschke products.

The functions have bounded Schwarzian derivatives.

They relate to univalent harmonic mappings.

Abstract

Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the condition $${\real} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2} ~\mbox{ for $z\in \ID$,} $$ where $0<\alpha\leq1$. The aim of this article is to show that this family has several elegant properties such as involving Blaschke products, Schwarzian derivative and univalent harmonic mappings.