On odd univalent harmonic mappings

Abstract

Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. In this article, we explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Motivated by the unresolved harmonic analogue of the Bieberbach conjecture, we investigate specific subclasses of \({\mathcal {S}}^0_H\), the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for functions exhibiting convexity in one direction and extend our findings to a more generalized class including the major geometric subclasses of \({\mathcal {S}}^0_H\). Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of p for which they belong. In particular, the results of this article enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We conclude the article with 2 conjectures and possible scope for further study as well.