Geometric subfamily of functions convex in some direction and Blaschke products

Abstract

Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h^{\prime }(0)=1$ and satisfying the condition $${\real} \left( \frac{z h''(z)}{αh'(z)}\right) <\frac{1}{2} ~\mbox{ for $z\in \ID$,} $$ where $0<\alpha \le 1$. 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.

arXiv comment12 pages; This article contains selected results of an unpublished version of a report of June 2020, but an enlarged version of it was circulated in 2020 and was also with a journal in June 2020