Harmonic pre-Schwarzian and its applications

The primary aim of this article is to extend the study on the pre-Schwarzian derivatives from the case of univalent analytic functions to that of univalent harmonic mappings defined on certain domains. This is done in two different ways. One of the ways is to connect with a conjecture on the univalent harmonic mappings. Also, we improve certain known results on the majorization of the Jacobian of functions in the affine and linear invariant family of sense-preserving harmonic mappings. This is achieved as an application of a corresponding distortion theorem in terms of the harmonic pre-Schwarzian derivative.