The Schwarzian Derivative of Harmonic Mappings in the Plane

Citation:

Liping NIE,Zongxin YANG.The Schwarzian Derivative of Harmonic Mappings in the Plane[J].Chinese Annals of Mathematics B,2020,41(2):193~208
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Authors:

Liping NIE; Zongxin YANG

Foundation:

This work was supported by the National Natural Science Foundation of China (No.11261022).
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 $\|S_{f}\|\leq6$. Furthermore, any locally univalent harmonic mapping $f$ which maps the unit disk onto an arbitrary regular $n$-gon has Schwarzian norm $\|S_{f}\|\leq\frac{8}{3}$.

Keywords:

Schwarzian derivative, Schwarzian norm, Harmonic mapping

Classification:

30C55, 30C62
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