On Koebe-type functions for harmonic quasiconformal mappings

TL;DR

This paper investigates a class of harmonic quasiconformal functions related to Koebe-type functions, providing new univalence criteria, coefficient bounds, and growth estimates that enhance existing mathematical understanding.

Contribution

It introduces new properties and inequalities for Koebe-type harmonic quasiconformal functions, extending prior results and deepening theoretical insights.

Findings

Established univalence conditions for the class

Derived coefficient inequalities and growth theorems

Improved upon previous results in harmonic quasiconformal mappings

Abstract

This paper studies a class of Koebe-type harmonic quasiconformal functions. It is motivated by the shear construction of Clunie and Sheil-Small [Ann. Acad. Sci. Fenn. Ser. A I Math. 9: 3--25, 1984] and the harmonic quasiconformal Koebe function. Equivalent univalence conditions, pre-Schwarzian and Schwarzian norms, coefficient inequalities, as well as growth and area theorems for this family of functions are established. These findings improve several previously known results.