The Schwarzian Derivative of Harmonic Mappings in the Plane

Abstract

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 ∥Sf∥ ≤ 6. Furthermore, any locally univalent harmonic mapping f which maps the unit disk onto an arbitrary regular n-gon has Schwarzian norm $$\left\| {{S_f}} \right\| \leq {8 \over 3}$$.