The Schwarzian Derivative of Harmonic Mappings in the Plane

In this paper, the authors introduce a definition of the Schwarzian derivative of any locally univalent harmonic mapping defined on a simply connected domain in the complex plane. Using the new definition, the authors prove that any harmonic mapping $f$ which maps the unit disk onto a convex domain has Schwarzian norm $\|S_{f}\|\leq6$. Furthermore, any locally univalent harmonic mapping $f$ which maps the unit disk onto an arbitrary regular $n$-gon has Schwarzian norm $\|S_{f}\|\leq\frac{8}{3}$.