## A factorization theorem for logharmonic mappings.(English)Zbl 1066.30020

A logharmonic mapping is a solution of a nonlinear elliptic partial differential equation $\overline{f}_{\overline{z}}= \left( a \frac {\overline{f}}{f} \right) f_z$ in a domain $$D$$ of $$C$$ where $$a$$ is an analytic function with $$| a|< 1$$ for all $$z\in D$$. The aim of the paper is a sufficient and necessary condition for a nonconstant logharmonic mapping $$f$$ to be factorized in the form $$f= F\circ \varphi$$ where $$\varphi$$ is analytic and $$F$$ is univalent logharmonic.

### MSC:

 30C62 Quasiconformal mappings in the complex plane

### Keywords:

logharmonic mapping; factorization
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